5(2 +2f)
U need to simplify this expression how do I get the answer and what is the answer pls help I’m so stuck on this
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Related Questions
Ect edt A weather center collects rain data for 10 years at two lakes. The amount of annual rainfall for each lake during that time period is shown by the graphs. Annual Rainfall Amounts Lake Warbler Lake Mockingbird + + + + + 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Rainfall (inches) 2 Which statement correctly describes the spreads of the data distributions for the lakes? The spreads of the data distributions are about the same for both lakes. BThe spread of the data distribution for Lake Mockingbird is about 40% greater than the /spread for Lake Warbler. The spread of the data distribution for Lake Mockingbird is 3 inches less than the spread of the data distribution for Lake Warbler. The spread of the data distribution for Lake Mockingbird is 5 inches greater than the spread of the data distribution at Lake Warbler. 3 Which statement is NOT true about the data? A The spreads for the first 25% of the data distributions for both lakes are equal. B The distribution for Lake Mockingbird shows a center that is 3 inches greater than the center of the distribution for Lake Warbler. C The interquartile range of the data distribution for Lake Warbler is about 60% of the interquartile range for Lake Mockingbird. D The data distributions for both lakes appear to be skewed left.
Answers
The correct answer is B. The spread of the data distribution for Lake Mockingbird is about 40% greater than the spread for Lake Warbler.
What is the data distribution?
The data distribution refers to the pattern of the data values in a dataset. It describes how the data is spread out, including the range of values, the shape of the distribution, and any clustering or gaps in the data. The data distribution can be described using various statistical measures, such as the mean, median, and standard deviation.
The correct answer to the first question is B: "The spread of the data distribution for Lake Mockingbird is about 40% greater than the spread for Lake Warbler."
For the second question, the statement that is NOT true about the data is B: "The distribution for Lake Mockingbird shows a center that is 3 inches greater than the center of the distribution for Lake Warbler."
hence, The correct answer is B. The spread of the data distribution for Lake Mockingbird is about 40% greater than the spread for Lake Warbler.
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which functions are symmetric with respect to the origin? y=arcsinx and y=arccosx
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Function[tex]y = arcsin(x)[/tex]is symmetric with respect to the origin, while function [tex]y = arccos(x)[/tex] is not. This is as inverse sine function is an odd function, while inverse cosine function is an even function.
If [tex]f(-x) = -f(x)[/tex] for all values of x in the function's domain, then the function is symmetric with regard to the origin. In other words, a function is said to be symmetric with regard to the origin if it passes through the origin and has the same shape on each side of the y-axis.
We must substitute -x for x in each function and simplify to find out if the functions [tex]y = arcsin(x)[/tex] and [tex]y = arccos(x)[/tex] are symmetric with respect to the origin.
According to the equation [tex]y = arcsin(x)[/tex], we get:
[tex]-arcsin = arcsin(-x) (x)[/tex]
Because it is odd, the function[tex]y = arcsin(x)[/tex] is symmetric with respect to the origin.
According to the equation [tex]y = arccos(x)[/tex]:
[tex]x(-arccos) = -arccos (x)[/tex]
Because it is not odd, the function[tex]y = arccos(x)[/tex] is not symmetric with respect to the origin.
In conclusion, [tex]y = arccos(x)[/tex]is not symmetric with regard to the origin, but [tex]y = arcsin(x[/tex] is. Due to the fact that the inverse cosine function is an even function, whereas the inverse sine function is an odd function, this is the case.
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. Translate parallelogram ABCD 2 units down and plot it. 2. Translate rhombus EFGH 2 units to the left, 4 units down, and plot it. 3. If the coordinates of the vertices of a square LMNO are L(-5,-5), M(-5,-2), N(-2,-2). What are the coordinates for O? 4. If the coordinates of the vertices of a triangle XYZ are X (2,1), Y(4,4) and Z(5,2). Translate the triangle XYZ 4 units down. What are the new coordinates? 5. Write the coordinates down for the following points:
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Answer:
To translate a shape, we need to move all its vertices by the same amount in the same direction. Here are the answers to the three parts of the question:
1. To translate parallelogram ABCD 2 units down, we need to subtract 2 from the y-coordinates of all its vertices. Let's say the original coordinates of the vertices are A(a,b), B(c,d), C(e,f), and D(g,h). Then the new coordinates of the vertices will be A'(a,b-2), B'(c,d-2), C'(e,f-2), and D'(g,h-2). Plot these new vertices to get the translated parallelogram.
2. To translate rhombus EFGH 2 units to the left and 4 units down, we need to subtract 2 from the x-coordinates and 4 from the y-coordinates of all its vertices. Let's say the original coordinates of the vertices are E(a,b), F(c,d), G(e,f), and H(g,h). Then the new coordinates of the vertices will be E'(a-2,b-4), F'(c-2,d-4), G'(e-2,f-4), and H'(g-2,h-4). Plot these new vertices to get the translated rhombus.
3. If the coordinates of the vertices of a square LMNO are L(-5,-5), M(-5,5), N(5,5), and O(5,-5), and we want to translate it 3 units to the right and 2 units up, we need to add 3 to the x-coordinates and subtract 2 from the y-coordinates of all its vertices. The new coordinates of the vertices will be L'(-2,-7), M'(-2,3), N'(8,3), and O'(8,-7). Plot these new vertices to get the translated square.
Use the polygon tool to graph the feasible region for the following constraints.
x+y≤6
2x+y≤8
x≥0
y≥0
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This allows you to visualize all of the possible solutions that meet the Constraint and determine the optimal solution for your problem.
To graph the feasible region for the constraint y≥0, you would start by using the polygon tool in a graphing program or software. This tool allows you to create and manipulate polygons on a coordinate plane.
To create the feasible region for the constraint y≥0, you would draw a horizontal line at y=0 on the coordinate plane. This line represents the boundary between the feasible and infeasible regions. Any point above the line is infeasible, while any point below the line is feasible.
Once you have drawn the horizontal line, you can use the polygon tool to create the feasible region by shading in the area below the line. This region represents all of the possible solutions that meet the constraint y≥0. Any point within this region would satisfy the constraint, while any point outside the region would violate the constraint.
Overall, using the polygon tool to graph the feasible region for the constraint y≥0 is a simple process that involves drawing a horizontal line at y=0 and shading in the area below it. This allows you to visualize all of the possible solutions that meet the constraint and determine the optimal solution for your problem.
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Help with math problems
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The value of the equation are 1. (-2√3 + 2)(√3 - 5) = -16 + 8√3 2. (5 - 4√5)(-2 + √5) = -30 + 21√. 3. (-2 - 3√5)(5 - √5) = -5 - 17√5. 4. (√5 - √3)(√5 + √3) = 2.
What are expressions and equations?An expression is a mathematical sentence without the equal sign that can include variables, operators, and integers. Although it can be appraised or simplified, there isn't a single perfect answer. An expression is, for instance, 3x + 2y.
A mathematical statement that has an equal sign and expresses the equivalence of two expressions is called an equation, on the other hand. To determine the values of variables that meet an equality, equations can be solved. One equation is 3x + 2y = 7, for instance.
1. Multiplying (-2√3 + 2)(√3 - 5), we get:
= -2√3 × √3 + (-2√3 × -5) + (2 × √3) + (2 × -5)
= -2√(3 × 3) + 10√3 - 2√3 - 10
= -2(3) + 8√3 - 10
= -16 + 8√3
Therefore, (-2√3 + 2)(√3 - 5) = -16 + 8√3.
2. Multiplying (5 - 4√5)(-2 + √5), we get:
= 5(-2) + 5√5 + (-4√5)(-2) + (-4√5)(√5)
= -10 + 13√5 + 8√5 - 4(5)
= -30 + 21√5
Therefore, (5 - 4√5)(-2 + √5) = -30 + 21√5.
3. Multiplying (-2 - 3√5)(5 - √5), we get:
= (-2 × 5) + (-2 × √5) + (-3√5 × 5) + (-3√5 × -√5)
= -10 - 2√5 - 15√5 + 3(5)
= -5 - 17√5
Therefore, (-2 - 3√5)(5 - √5) = -5 - 17√5.
4. Multiplying (√5 - √3)(√5 + √3), we get:
= (√5 × √5) - (√3 × √3)
= 5 - 3
= 2
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A shopkeeper had 4 handbags which were of the same cost price. He sold 3 of them at 40% more than the cost price. He sold the fourth handbag at cost price. He received $260 altogether. Find the cost price of each handbag.
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Change numbers from one base to another. Perform base conversions using multiplication expansion and divisions, and distinguish the differences between the two through writing.
Through this activity, you see how the hexadecimal value of color is represented in HTML and practice how to convert among hexadecimal, binary, and decimal values.
Conner took a graphic design class and learned:
Combining red, green, and blue (RGB) light is the standard method of producing color images on screens. In HTML code, the color red is expressed as a hexadecimal #FF 00 00 or a 24-bit color depth (bit is binary digit), which means the browser is showing only red (FF) with no green (00) and no blue (00).
Now Conner got this question on the Liberal Arts Math class:
Q: What is FF equivalent to in Decimals and what is the equivalent in Binary?
He was so confident in his answers since he already knew the letter F is 1111 from his graphic design class.
Below shows how Conner answered the question, but Conner didn't receive full credit. He argued that he was confident with the solution, and also his numbers were the same as the answer key. (Answer key: 255 and 11111111)
Conner's incorrect answer:
#FF0000 converted to base ten (decimal) is: 255 (Incorrect)
#FF0000 converted to base two (binary) is: 11111111 (Incorrect)
1. Conner didn't show the details of the conversions, but that is not the reason why he lost points. Please explain to him why his answer didn't receive full credit. Note: It is not because he didn't show work. Help Conner correct his mistakes. Please show details of the conversions.
2. Show how to convert 255 to the binary 11111111 using divisions. Please show the detailed steps. Be specific with your answer. If work is done on paper, please attach the image file.
3. Describe how you convert a decimal number (base ten) to another base using divisions. Use your own words to provide an instructional description.
Answers
Answer:
137 in binary is 10001001
Step-by-step explanation:
Conner's answer did not receive full credit because he did not convert the hexadecimal number #FF0000 to decimal and binary correctly.
While it is true that the letter F represents 1111 in binary, in hexadecimal, FF represents the decimal value 255, not 1111.
Therefore, Conner's answer is incorrect, and he needs to use the correct conversions to get the right answer.
To convert 255 to binary using divisions, we can use the following steps:
Start with the decimal number 255.
Divide 255 by 2, and write down the quotient and remainder. The quotient is 127, and the remainder is 1.
Divide the quotient (127) by 2, and write down the quotient and remainder. The quotient is 63, and the remainder is 1.
Repeat this process of dividing the quotient by 2 and writing down the quotient and remainder until the quotient becomes 0.
The binary equivalent of 255 is the sequence of remainders written in reverse order: 11111111.
Decimal Number Quotient Remainder
255 127 1
127 63 1
63 31 1
31 15 1
15 7 1
7 3 1
3 1 1
1 0 1
Divide 137 by 2. The quotient is 68, and the remainder is 1.
Divide 68 by 2. The quotient is 34, and the remainder is 0.
Divide 34 by 2. The quotient is 17, and the remainder is 0.
Divide 17 by 2. The quotient is 8, and the remainder is 1.
Divide 8 by 2. The quotient is 4, and the remainder is 0.
Divide 4 by 2. The quotient is 2, and the remainder is 0.
Divide 2 by 2. The quotient is 1, and the remainder is 0.
Divide 1 by 2. The quotient is 0, and the remainder is 1.
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Can someone please help me? I have to put this equation in order. I already tried but I was missing something.
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I don't think anything is wrong with your values. Since it's an equation, you can see something different and get the right answer.
Someone help me with this pls!!!
Answers
Answer:
8. A) 3.75
9. B) 5.00
You are fencing in a rectangular area of a garden you have only 150 feet of fence do you want the length of the garden to be at least 40 feet you want the width of the garden to be at least 5 feet what is a graph showing the possible dimensions your garden could have? What vegetables will you use? What will they represent? How many inequalities do you need to write?
Answers
Answer:
Length ≥ 40
Width ≥ 5
Perimeter = 2 × (Length + Width)
2 × (Length + Width) ≤ 150
Step-by-step explanation:
To create a graph showing the possible dimensions of the garden, we need to plot the length and width of the rectangular area on the x and y axes, respectively. Since we want the length to be at least 40 feet and the width to be at least 5 feet, we can represent these constraints by the following inequalities:
Length ≥ 40
Width ≥ 5
We also know that the total length of fencing available is 150 feet, which means that the perimeter of the rectangular area must be less than or equal to 150 feet. The perimeter of a rectangle is given by:
Perimeter = 2 × (Length + Width)
So, we can write the inequality representing the perimeter as:
2 × (Length + Width) ≤ 150
To graph the possible dimensions of the garden, we can plot the points that satisfy all three inequalities on the x-y plane.
Regarding the vegetables, it is not clear what vegetables the user would like to plant in the garden. As such, we cannot provide a specific answer to this question.
In summary, we need to write three inequalities to represent the constraints in the problem, and we can graph the solution space using these inequalities.
Write the point-slope form of the line satisfying the given conditions. Then use the point-slope form of the equation to write the slope-intercept form of the equation.
Slope = 4, passing through (-7,6)
Type the point-slope form of the equation of the line.
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
Type the slope-intercept form of the equation of the line.
(Use integers or simplified fractions for any numbers in the equation.)
Answers
Answer:
y = 7x + 10
Step-by-step explanation:
Use your point-slope form: y- y1 = m (x - x1) to substitute the given information.
y - 6 = 7(x - (-4))
Then simplify.
y - 6 = 7(x + 4)
Next, distribute the 7 into the parenthesis which will give you the point-slope form.
y - 6 = 7x + 28
In order to write it in slope-intercept form, add 6 to both sides.
y - 6 + 6 = 7x + 4 + 6
Then simplify to write the equation in slope-intercept form.
y = 7x + 10
To write the point-slope form of the equation of a line, we use the formula:
y - y1 = m(x - x1)
where m is the slope of the line and (x1, y1) is a point on the line.
Given that the slope of the line is 4 and it passes through the point (-7, 6), we can substitute these values into the formula to get:
y - 6 = 4(x - (-7))
Simplifying this equation, we get:
y - 6 = 4x + 28
To write the slope-intercept form of the equation, we need to solve for y:
y = 4x + 34
Therefore, the point-slope form of the equation of the line is y - 6 = 4(x + 7), and the slope-intercept form of the equation of the line is y = 4x + 34.
Luther Williams’ charge account statement shows an unpaid balance of $3,987.11. The monthly finance charge is 1.75 percent of the unpaid balance. What is the new account balance? Show your work.
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By adding the monthly finance charge to the unpaid balance, the new account balance is $4,056.91.
What is addition?
Addition is a mathematical operation that involves combining two or more numbers to produce a sum or total. It is commonly denoted by the plus sign (+) and is one of the four basic arithmetic operations.
To find the new account balance, we need to add the monthly finance charge to the unpaid balance.
First, we need to calculate the monthly finance charge:
Monthly finance charge = 1.75% of unpaid balance
= 0.0175 * $3,987.11
= $69.80
Next, we add the monthly finance charge to the unpaid balance:
New account balance = Unpaid balance + Monthly finance charge
= $3,987.11 + $69.80
= $4,056.91
Therefore, the new account balance is $4,056.91.
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Roger is 5’11”. How much should he weigh in order to have a BMI of 25?
Answers
Because it’s logic
Per your formula, B=(705w)/h2, where w=weight in pounds and h=height in inches.
From the info provided, we know that B=25 and h=74 inches (6ft * 12 inches per foot + 2 inches).
Substituting these values in the formula, we get: 25=(705w)/(74*74)
25=705w/5476
Now, solving for w, we get w=(25*5476)/705=194.18 pounds (rounded to nearest hundredth).
Hope this helps.
On Monday, Tuesday, and Wednesday, Cheryl ran 1 miles each day. On Thursday, she ran 2 miles. On Friday,
she ran 3 miles.
How many miles, in total, did Cheryl run for those five days?
Write your answer as a decimal.
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Cheryl ran a total of 8 miles in five days.
1 + 1 + 1 + 2 + 3 = 8
Written as a decimal, this is 8.0 miles.
Which number line shows ï > 7? A B 0 1 2 3 4 5 6 7 8 9 10 * 0 1 2 3 4 5 6 7 8 9 10 OF A → B Submit
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Answer:
The number line that shows ï > 7 would be from 8 to 10, as the symbol > means greater than. So the correct answer is B. Submit B.
forever quilting is a small company that makes quilting kits priced at $120 each. there is no quantity discount. the costs of the materials that go into each kit total $45. it costs $5 in labor to assemble a kit. the company has monthly expenses of $1,000 for rent and insurance, $200 for heat and electricity, $500 for advertising, and $4,500 for the monthly salary of its owner. last month the company sold 150 kits in a given month, calculate it’s monthly profits
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Forever Quilting's monthly profit is $10,250. We can caclculate it in the following manner.
To calculate the company's monthly profit, we need to subtract its total expenses from its total revenue.
Total Revenue = Number of kits sold x Price per kit
Total Revenue = 150 kits x $120 per kit
Total Revenue = $18,000
Total Costs = (Cost of materials per kit + Labor cost per kit) x Number of kits sold + Monthly expenses
Total Costs = ($45 + $5) x 150 kits + $1,000 + $200 + $500 + $4,500
Total Costs = $7,750
Monthly Profit = Total Revenue - Total Costs
Monthly Profit = $18,000 - $7,750
Monthly Profit = $10,250
Therefore, Forever Quilting's monthly profit is $10,250.
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Candice makes bracelets that contain 2 oval-shaped beads and 8 circular-shaped beads.
She has 32 oval-shaped beads and 38 circular-shaped beads.
How many bracelets can she make?
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She has 32 oval-shaped beads, which means she can make 32/2 = 16 bracelets using only the oval-shaped beads.
She also has 38 circular-shaped beads, which means she can make 38/8 = 4.75 bracelets using only the circular-shaped beads.
Since Candice cannot make a fractional number of bracelets, the maximum number of bracelets she can make is limited by the number of circular-shaped beads she has. Therefore, she can only make 4 bracelets using the circular-shaped beads.
So, Candice can make a total of 20 bracelets.
Lulu observes that when she dilates a quadrilateral from a center of dilation outside the
quadrilateral, the corresponding sides of the original figure and its dilated image are
parallel. Which statement best explains why this occurs?
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As a result, the original figure's corresponding sides and those of its enlarged image are parallel.
what is quadrilateral ?A quadrilateral is a closed, two-dimensional form in geometry that has four straight sides. Four sides, four edges, and four angles make up the polygonal shape. The total of a quadrilateral's internal angles is 360 degrees. Kites, squares, rectangles, parallelograms, and trapezoids are examples of quadrilaterals.
given
Each side of the original figure is expanded to meet the dilation scale factor when a quadrilateral is dilated from an outer centre of dilation, and the sides of the dilated image are parallel.
Parallel lines continue to be parallel after scaling because dilatation scales each side of the quadrilateral by a constant factor.
As a result, the original figure's corresponding sides and those of its enlarged image are parallel.
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The line of its elements from the upper left corner to the lower right of the second order determinant is called?
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The line of its elements from the upper left corner to the lower right of the second order determinant is called the main diagonal.
Simplify |x/3| if x>0
Answers
Answer: -31x
Step-by-step explanation:
|x|31 becomes -x31 which is -31x
Jennifer pours 1/3 quart of milk equally into 4 glasses. How much milk, in quarts, does Jennifer pour into each glass?
Answers
Answer:
1 ⅓
Step-by-step explanation:
there 4 glasses and you put ⅓ in each one therefore 4 x ⅓ = 1 ⅓
The Bells obtain a 30 year $110,000 conventional mortgage at 10% on a house selling for $130,000 their monthly mortgage payment including principal and interest is $957 determine the total amount they will pay for their house 
Answers
Answer:
what is 5+17894874298748
Answer:
Therefore, the total amount the Bells will pay for their house is $344,520.
Step-by-step explanation:
We can start by using the information about the monthly mortgage payment to calculate the total number of payments they will make over the 30-year term:
30 years × 12 months/year = 360 months
So the Bells will make 360 monthly payments of $957. To find the total amount they will pay for the house, we need to multiply the monthly payment by the number of payments:
$957/month × 360 months = $344,520
Therefore, the total amount the Bells will pay for their house is $344,520.
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Solve the systems by elimination.
2x + 4y = -16
-2x + 2y = -2
Answers
Answer: (-2,-3)
Step-by-step explanation:
A line segment is shown on the coordinate grid.
Select all of the numbers that are included within the range of the line segment.
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Without the specific details or an image of the line segment on the coordinate grid, I'm unable to provide you with the specific numbers included in the range.
To help you generally, a line segment is a part of a line with two endpoints, and it is located on a coordinate grid, which is a two-dimensional plane formed by the intersection of horizontal and vertical lines (x and y axes).
To determine the range of the line segment, you need to identify the y-coordinates of the two endpoints.
The range includes all the y-coordinates between those two points, inclusive of the endpoints. To find the range, observe the line segment and note the y-coordinates of both endpoints. Then, list all the numbers (whole or fractional) between those values, including the endpoints themselves.
Once you have the necessary details or an image of the line segment, feel free to ask again, and I'll be happy to help you identify the numbers within the range.
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Let S be the set R^2. Define addition and multiplication operations on S as follows: for all real numbers a,b,c,d,
(a,b)+(c,d) := (a+c,b+d), (a,b)·(c,d):=(bd−ad−bc,ac−ad−bc).
(a) Prove the right distributive law for S.
(b) What is the multiplicative identity element for S? Explain how you found it. (c)Using (b), prove the multiplicative identity law for S.
Answers
(a) The right distributive law is proven for S.For S, this is expressed as (a,b)·(c,d)=(a,b)·(c,d)=(a·c,a·d+b·c).
(a,b)·(c,d)=(bd−ad−bc,ac−ad−bc)=(a·c,a·d+b·c).
(b) we can deduce that x=1 and y=0. Hence, the multiplicative identity element for S is (1,0).
(c) For S, this is expressed as (a,b)·(1,0)=(a,b). Let (a,b) be an arbitrary element in S. The multiplicative identity law is proven for S.
What is distributive law?The right distributive law states that the product of a scalar and a vector is equal to the sum of the products of the scalar with each component of the vector.
(a) For S, this is expressed as (a,b)·(c,d)=(a,b)·(c,d)=(a·c,a·d+b·c).
To prove this, we need to show that (a,b)·(c,d)=(a·c,a·d+b·c).
Let (a,b) and (c,d) be two arbitrary elements in S. Then,
(a,b)·(c,d)=(bd−ad−bc,ac−ad−bc)
=(a·c,a·d+b·c).
Therefore, the right distributive law is proven for S.
(b) The multiplicative identity element for S is (1,0). To find this, we need to find a pair of elements (x,y) such that (a,b)·(x,y)=(a,b).
Let (x,y) be an arbitrary pair of elements in S. Then,
(a,b)·(x,y)=(ax−by,ay+bx)
=(a,b).
From this, we can deduce that x=1 and y=0. Hence, the multiplicative identity element for S is (1,0).
(c) The multiplicative identity law states that the product of a scalar and the multiplicative identity element is equal to the scalar itself. For S, this is expressed as (a,b)·(1,0)=(a,b).
To prove this, we need to show that (a,b)·(1,0)=(a,b).
Let (a,b) be an arbitrary element in S. Then,
(a,b)·(1,0)=(a·1−b·0,a·0+b·1)
=(a,b).
Therefore, the multiplicative identity law is proven for S.
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The product of eight and two, minus the product of three and four
Answers
Answer:
4
Step-by-step explanation:
The product of eight and two is 8 x 2 = 16.
The product of three and four is 3 x 4 = 12.
So, the expression "the product of eight and two, minus the product of three and four" can be written as:
16 - 12 = 4
Therefore, the answer is 4.
Victoria is 5 years older than her brother Elliot. Let k represent Elliot's age. Identify the expression that can be used to find Victoria's age.
Answers
Answer:
k+5
Step-by-step explanation:
k= Elliot
Victoria is 5 years older than Elliot
=k+5
The value of an industrial embroidery machine is decreasing according to the function defined by:
V(t) = 11,700(3)^-0.15
Where t is the number of years since the machine was purchased. What does the y-intercept represent?
A. The value of the machine after 3 years
B. The amount of time it takes for the value of the machine to reach zero
C. The rate at which the value of the machine depreciates
D. The original value of the machine
I think it is A. The value of the machine after 3 years
What do you think?
Answers
V(0) = 11,700(3)^-0.15
Simplifying this expression, we get:
V(0) = 11,700(1)
V(0) = 11,700
Therefore, the y-intercept of the function is 11,700. This means that when the machine was purchased (i.e., at t = 0), its value was $11,700.
So, the answer is D. The y-intercept represents the original value of the machine.
Find all integer solutions of
x+y=3
3xy-z*z=9
Answers
In conclusion, the integer solutions of the given system of equations are [tex]$(x, y, z) = (1, 2, 0)$ and $(x, y, z) = (-1, 4, 0)$.[/tex]
What is the integer?An integer is a positive, negative, or zero-valued whole number. It is a number without a fractional or decimal component.
To find all integer solutions of the given system of equations:
[tex]x + y &= 3 \[/tex]
[tex]3xy - z^2 &= 9[/tex]
We can start by solving the first equation for one of the variables. Let's solve for y in terms of x:
[tex]x + y &= 3 \[/tex]
[tex]y &= 3 - x[/tex]
Now we substitute this expression for y into the second equation:
[tex]3xy - z^2 &= 9 \[/tex]
[tex]3x(3-x) - z^2 &= 9 \quad \text{Substituting } y = 3-x \[/tex]
[tex]9x - 3x^2 - z^2 &= 9 \[/tex]
[tex]3x^2 + z^2 &= 9x - 9 \[/tex]
[tex]3x^2 - 9x + z^2 &= -9 \quad \text{(1)}[/tex]
Now we will examine equation (1) for integer solutions.
Case 1: [tex]z= 0[/tex]
If then equation (1) becomes:
[tex]3x^2 - 9x &= -9 \quad \text{(1a)} \[/tex]
[tex]3x(x-3) &= -9[/tex]
Since we are looking for integer solutions, 3x and x-3 must have opposite signs. The possible pairs of factors of -9 with opposite signs are (3,-3) and (-3,3). So we have two possible sets of equations to solve:
The discriminant is negative, there are no integer solutions. Therefore, there are no integer solutions for the original system of equations when [tex]z \neq 0.[/tex]
In conclusion, the integer solutions of the given system of equations are [tex]$(x, y, z) = (1, 2, 0)$ and $(x, y, z) = (-1, 4, 0)$.[/tex]
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The time taken for healthy Canadian adults to complete a logic problem is believed to have a mean 40 seconds. It is of interest to investigate whether UBC students perform better on average than healthy adult Canadians, so the logic problem is given to a sample of 80 UBC students, and their times to solution are recorded. The sample mean and standard deviation are 36 seconds and 17 seconds.
Part a) What is/are the parameters of interest relevant to this hypothesis test? Choose all parameters that you use to set up the null and alternative hypotheses, as well as those referenced in the assumptions and derivation of the relevant test statistic. Hint: A value (number) by itself is not a parameter.
A. 80
B. 40 seconds
C. The mean time for the 80 UBC students to complete the logic problem.
D. The mean time for all UBC students to complete the logic problem.
E. None of the above
Part b) In testing a hypothesis about a parameter of interest, what would your null hypothesis be?
The mean time taken to solve the logic problem by healthy Canadian adults is 40 seconds.
The mean time taken to solve the logic problem by healthy Canadian adults is greater than 40 seconds.
The mean time taken to solve the logic problem by healthy Canadian adults is less than 40 seconds.
The mean time taken to solve the logic problem by healthy Canadian adults is different from 40 seconds.
The mean time taken to solve the logic problem by UBC students is greater than 40 seconds.
The mean time taken to solve the logic problem by UBC students is less than 40 seconds.
The mean time taken to solve the logic problem by UBC students is different from 40 seconds.
The mean time taken to solve the logic problem by UBC students is 40 seconds.
Part c) You would take the alternative hypothesis to be:
one-sided, left-tailed
two-sided.
one-sided, right-tailed.
it does not matter whether we take a one-sided or two-sided alternative.
Part d) Compute the test statistic (Please round your answer to three decimal places):
Part e) Assume all necessary conditions are met (random sampling, independence samples, large enough sample size). Which of the following approximate the sampling distribution of the test statistic in Part d:
Normal distribution
t-distribution
Part f) Suppose that, based on data collected, you reject the null hypothesis. Which of the following could you conclude? Note: Read these carefully. I know they all sound the same, but they are all saying different things.
There is sufficient evidence to suggest the mean time taken to solve the logic problem by UBC students is less than the mean time for healthy adult Canadians.
There is sufficient evidence to suggest the mean time taken to solve the logic problem by UBC students is the same as the mean time for healthy adult Canadians.
There is sufficient evidence to suggest the mean time taken to solve the logic problem by UBC students is greater than the mean time for healthy adult Canadians.
There is insufficient evidence to suggest the mean time taken to solve the logic problem by UBC students is the same as the mean time for healthy adult Canadians.
There is insufficient evidence to suggest the mean time taken to solve the logic problem by UBC students is less than the mean time for healthy adult Canadians.
There is insufficient evidence to suggest the mean time taken to solve the logic problem by UBC students is greater than the mean time for healthy adult Canadians.
Part g) Suppose that, based on data collected, you decide that UBC students perform better on average than healthy adult Canadians. Note: Read these carefully. I know they all sound the same, but they are all saying different things.
it is possible that you are making a Type I error.
it is possible that you are making a Type II error.
it is certainly correct that UBC students perform better on average than healthy adult Canadians.
it is certainly incorrect that UBC students perform better on average than healthy adult Canadians.
there must have been a problem with the way the sample was obtained.
Part h) Suppose that, based on the data collected, you obtain a P
-value of 0.02 (confirm this using the t-table). This means:
the sample of UBC students performed relatively better, if indeed the true mean time taken to solve the logic problem by all UBC students is 40 seconds.
there is a 2% chance that UBC students perform better on average than healthy adult Canadians.
there is a 2% chance that UBC students perform worse on average than healthy adult Canadians.
the probability of UBC students performing as well or better is 0.02, if indeed the true mean time taken to solve the logic problem by all UBC students is 40 seconds.
the probability of UBC students performing as well or worse is 0.02, if indeed the true mean time taken to solve the logic problem by all UBC students is 40 seconds.
the sample of UBC students performed relatively worse, if indeed the true mean time taken to solve the logic problem by all UBC students is 40 seconds.
Answers
a) 40 seconds is hypothesized mean b) mean time to solve problem by UBC students = 40 sec is null hypothesis c) Alternative hypothesis is time taken is different from 40 sec d) t = -2.353 h) 2% sample mean
Part a) The following variables are significant to this hypothesis test:
B. The null hypothesis is based on the assumption that it will take healthy Canadian adults an average of 40 seconds to solve the logic puzzle.
D. The test statistic is calculated using the sample mean, which is the average time it took the 80 UBC students to solve the logic puzzle.
D. Based on our sample data, we wish to determine the mean time it takes for all UBC students to complete the logic problem.
Part b) The null hypothesis would be that it took UBC students an average of 40 seconds to solve the logic puzzle.
The alternate hypothesis (part c) is that students at UBC take longer than 40 seconds on average to answer the logic puzzle (two-sided alternative).
Part d) You can compute the test statistic as:
t = [tex]\sqrt{sample size} / (sample standard deviation / hypothesised mean) / (sample mean - hypothesised mean)[/tex]
[tex]t = (36 - 40) / (17 / \sqrt{80} )[/tex]
t = -2.353
Part e) The normal distribution can be used to approximate the sampling distribution of the test statistic because the sample size is high (n = 80).
Part f) If we reject the null hypothesis, we can draw the conclusion that there is enough evidence to support the idea that UBC students took longer on average than healthy adult Canadians to solve the logic puzzle.
Part g) It's feasible that we are committing a Type I error if we determine that UBC students perform on average better than healthy adult Canadians (rejecting the null hypothesis when it is actually true).
Part h) If the null hypothesis is correct, a p-value of 0.02 indicates that there is a 0.02 chance of receiving a sample mean that is as severe as the one we observed (or more extreme). In other words, there is only a 2% probability of observing a sample mean that is as different from 40 seconds (or more different) as the one we obtained, if the true mean time taken to solve the logic problem by all UBC students is 40 seconds. Hence, at a significance level of 0.05 but not at a significance level of 0.01 we would reject the null hypothesis.
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