Using the given operations and nine 9s, you can make 1001.
How to make 1001 from nine 9. Use can use 9 in any wayTo make 1001 using nine 9s, you can use the following mathematical operations:
1. (9 + 9 + 9) * 9 * 9 - 9 - 9 - 9 = 1001
- Adding three 9s together: (9 + 9 + 9) = 27
- Multiplying the sum by two 9s: (27 * 9 * 9) = 2187
- Subtracting three 9s: (2187 - 9 - 9 - 9) = 2160
- Adding one 9: (2160 + 9) = 2169
- Adding 832 (which is (9 * 9 * 9 + 9 * 9 + 9)) to 2169: (2169 + 832) = 3001
- Subtracting two 9s: (3001 - 9 - 9) = 2983
- Adding nine 9s: (2983 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9) = 1001
Therefore, using the given operations and nine 9s, you can make 1001.
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Which is the graph of the linear inequality 1/2x – 2y > –6? On a coordinate plane, a solid straight line has a positive slope and goes through (negative 4, 2) and (4, 4). Everything above and to the left of the line is shaded. On a coordinate plane, a dashed straight line has a positive slope and goes through (negative 4, 2) and (4, 4). Everything above and to the left of the line is shaded. On a coordinate plane, a solid straight line has a positive slope and goes through (negative 4, 2) and (4, 4). Everything below and to the right of the line is shaded. On a coordinate plane, a dashed straight line has a positive slope and goes through (negative 4, 2) and (4, 4). Everything below and to the right of the line is shaded.
The correct graph of the linear inequality 1/2x - 2y > -6 is the one where a solid straight line has a positive slope and goes through (negative 4, 2) and (4, 4), and everything below and to the right of the line is shaded.
HELP PLESSE
The total cost of a lunch is shared among 8 people. the total bill is 55 what is the cost
Answer: A,
Step-by-step explanation:
8 people, times whatever each person payed will equal to 55$ in total
Which graph represents a function
Answer:
The one at the bottom right above the next button
Step-by-step explanation:
Suppose the bear population in the Allegheny National Forest have weights that produce a normal density curve as shown.
74——————-200————————-326
From the graph shown, use the 69-95-99.7% (empirical) rule to estimate the standard deviation of the bear weights.
The estimated standard deviation of the bear weights, based on the given graph and using the 69-95-99.7% rule, is approximately 63.
The 69-95-99.7% (empirical) rule, also known as the 3-sigma rule, is a rule of thumb that applies to data that follows a normal distribution. According to this rule:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
In the given graph, if we assume the bear weights follow a normal distribution, we can estimate the standard deviation using the 69-95-99.7% rule.
Based on the graph, we know that the midpoint of the distribution (mean) is 200. Assuming the graph is symmetric, we can estimate one standard deviation as half the distance between the mean (200) and either end (74 or 326).
To calculate this, we subtract the mean from one of the endpoints and divide by 2:
Standard Deviation ≈ (326 - 200) / 2 ≈ 126 / 2 ≈ 63
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Choose an amount between $60.00 and $70.00 to represent the cost of a grocery bill for a family. Be sure to include dollars and cents.
Part A: If the family has a 25% off coupon, calculate the new price of the bill. Show all work or explain your steps. (6 points)
Part B: Calculate a 7% tax using the new price. What is the final cost of the bill? Show all work or explain your steps. (6 points)
Answer:
A: $51.00
B: $54.57
Step-by-step explanation:
Let amount = $68.00
Part A:
Since the coupon is for 25%, the family pays 75% of $68.00
75% of $68.00 = 0.75 × $68.00 = $51.00
The new price is $51.00
Part B:
The tax is 7% of $51.00
7% of $51.00 = $3.57
The total price is the sum of $51.00 and the amount of tax, $3.57
Total price = $51.00 + $3.57 = $54.57
QUESTION 3 Find the value of x in the figure below. (4 marks) a) (5x15) +5+45- 45°
The calculated value of x in the triangle is 40°
How to calculate the value of xFrom the question, we have the following parameters that can be used in our computation:
The figure (see attachment)
Using the theorem of linear pair, we have
∠DBA + ∠ABC = 180°
Using the given values, we have
∠100° + ∠ABC = 180°
Collect the like terms
∠ABC = 180° - 100°
Evaluate
∠ABC = 80°
The sum of angles of a triangle is 180° .
So, in triangle ABC
∠A + ∠B + ∠C = 180°
Using the given values, we have
x + 80 + 60 = 180°
Evaluate the sum
x + 140 = 180°
Collect the like terms
x = 180° - 140°
Evaluate
x = 40°
Hence, the value of x is 40°
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Determine algebraically, the solution interval for the quadratic inequality 2x²-7x≤-3
Interval
Test Point
Substitution
True/False?
Solution:
Answer:
violence figer in the past two years
Which of the following lists of ordered pairs is a function?
The list of ordered pairs that is a function is Option D.
What is a Math Function?A math function is a relationship that assigns a unique output value to each input value. It describes how one quantity depends on another.
Functions are commonly represented using mathematical notation, such as f(x), and they play a fundamental role in various areas of mathematics and its applications.
A function is a relation in which one input (x-value) is assigned to exactly one output (y-value).
Since option D's x-values do not repeat, then it is a function.
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determine the value of x
Answer:
[tex]x = 5\sqrt3[/tex]
Step-by-step explanation:
We can solve for the side length x in this 30-60-90 triangle by using the ratio of side lengths for that specific type of right triangle:
1 : [tex]\sqrt3[/tex] : 2In this triangle, we can identify the smallest side (corresponding to 1 in the ratio) as 5. This means we can solve for x by multiplying 5 by [tex]\sqrt3[/tex]. Thus:
[tex]\boxed{x = 5\sqrt3}[/tex]
ily sold 18 items at the street fair. She sold bracelets for $6 each and necklaces for $5 each for a total of $101. Which system of equations can be used to find b, the number of bracelets she sold, and n, the number of necklaces she sold?
b + n = 101
6b + 5n = 18
b + n = 101
5b + 6n = 18
b + n = 18
6b + 5n = 101
b + n = 18
5b + 6n = 101
Answer:
6b + 5n = 101
Step-by-step explanation:
The correct system of equations that can be used to find b, the number of bracelets Ily sold, and n, the number of necklaces she sold is:
b + n = 18
6b + 5n = 101
In this system, the first equation represents the total number of items sold, which is 18. Since b represents the number of bracelets and n represents the number of necklaces, the equation b + n = 18 reflects that the total number of bracelets and necklaces sold should add up to 18.
The second equation represents the total amount of money Ily earned from selling bracelets and necklaces. Since bracelets were sold for $6 each and necklaces for $5 each, the equation 6b + 5n = 101 represents the total amount of money earned, which is $101.
Therefore, the correct system of equations is:
b + n = 18
6b + 5n = 101
The table shows the size of outdoor decks (x) in square feet, and the estimated dollar cost to construct them (y).
x y x2 xy
100 600 10,000 60,000
144 850 20,736 122,400
225 1,300 50,625 292,500
324 1,900 104,976 615,600
400 2,300 160,000 920,000
∑x=1,193 ∑y=6,950 ∑x2=346,337 ∑xy=2,010,500
Which regression equation correctly models the data?
y = 5.83x – 1.04
y = 5.83x + 17
y = 5.71x + 29
y = 5.71x + 27.6
The regression equation that correctly models the data is: y = 5.71x + 27.6.
The correct answer to the given question is option D.
Regression equations are mathematical models that relate two or more variables to find the relationship between them. One variable, denoted as y, is considered the dependent variable. The other variable, denoted as x, is considered the independent variable.
In this case, the independent variable is the size of the outdoor deck, while the dependent variable is the estimated cost to construct it.
There are different types of regression equations. The one that fits this scenario is the linear regression equation, which has the form y = mx + b, where m is the slope of the line and b is the y-intercept.
The slope represents the change in y for each unit change in x, while the y-intercept represents the value of y when x is zero. To find the regression equation that correctly models the data, we need to calculate the slope and the y-intercept using the given values.
We can use the following formulas:
Slope: m = [(n∑xy) - (∑x)(∑y)] / [(n∑x2) - (∑x)2]
Y-intercept: b = (∑y - m∑x) / n Where n is the number of data points, which is 6 in this case.
Using the given values, we get: Slope: m = [(6)(2,010,500) - (1,193)(6,950)] / [(6)(346,337) - (1,193)2] = 5.71
Y-intercept: b = (6,950 - (5.71)(1,193)) / 6 = 27.6
Therefore, the regression equation that correctly models the data is: y = 5.71x + 27.6
The answer is option D.
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Maricella solves for x in the equation 4 x minus 2 (3 x minus 4) + 4 = negative x + 3 (x + 1) + 1. She begins by adding –4 + 4 on the left side of the equation and 1 + 1 on the right side of the equation. Which best explains why Maricella’s strategy is incorrect?
A. The multiplication that takes place while distributing comes before addition and subtraction in order of operations.
B. In order to combine like terms on one side of the equation, the inverse operation must be used.
C. When the problem is worked in the correct order, the numbers that Maricella added are not actually like terms.
D. Maricella did not combine all three constants on both sides of the equation; she combined only two.
Answer:
A. The multiplication that takes place while distributing comes before addition and subtraction in order of operations.
Step-by-step explanation:
You want to know the error that adding -4+4 on the left and 1+1 on the right represents in the solution of the equation ...
4x -2(3x -4) +4 = -x +3(x +1) +1SolutionThe correct solution procedure would be to eliminate the parentheses using the distributive property as a first step:
4x -6x +8 +4 = -x +3x +3 +1
Comparing this to Maricella's first step, we see that she ignored the step of using the distributive property to multiply the constants inside parentheses by the factor outside. The appropriate description of Maricella's mistake is ...
A. The multiplication that takes place while distributing comes before addition and subtraction in order of operations.
__
Additional comment
Adding like terms would give ...
-2x +12 = 2x +4
8 = 4x . . . . . . . . . . . add 2x-4 to both sides
2 = x . . . . . . . . . . divide by 4
<95141404393>
The results of an analysis, on the makeup of garbage, done by the Environmental Protection Agency was published in
1990. Some of the results are given in the following table, which for various years gives the number of pounds per
person per day of various types of waste materials.
Waste materials
Glass
Plastics
Metals
Paper
1960
0.20
0.01
0.32
0.91
1970
0.34
0.08
0.38
1.19
1980
0.36
0.19
0.35
1.32
1988
0.28
0.32
0.34
1.60
For metal, calculate the average rate of change between 1980 and 1988. Then interpret what this value means.
a. From 1980 to 1988, the number of pounds of c. From 1980 to 1988, the number of pounds of
metal per person per day decreased by
metal per person per day decreased by
0.125 per year.
0.00125 per year.
b. From 1980 to 1988, the number of pounds d. From 1980 to 1988, the number of pounds
of metal per person per day decreased by
0.071 per year.
of metal per person per day increased by
0.01 per year.
The average rate of change for the number of pounds of metal per person per day between 1980 and 1988 is -0.00125 pounds per year.
To calculate the average rate of change for the number of pounds of metal per person per day between 1980 and 1988, we need to find the difference in the values and divide it by the number of years.
In 1980, the pounds of metal per person per day was 0.35, and in 1988, it was 0.34. The difference between these values is -0.01.
The number of years between 1980 and 1988 is 1988 - 1980 = 8 years.
Now, we can calculate the average rate of change:
Average rate of change = (Change in pounds of metal) / (Number of years)
= (-0.01) / 8
= -0.00125
The average rate of change for the number of pounds of metal per person per day between 1980 and 1988 is -0.00125 pounds per year.
Interpretation:
The negative value of the average rate of change (-0.00125) indicates that there was a decrease in the number of pounds of metal per person per day from 1980 to 1988.
Specifically, on average, there was a decrease of approximately 0.00125 pounds per year.
This suggests that there was a declining trend in the use or disposal of metal waste during this period.
It could indicate improvements in recycling or waste management practices, or a shift in consumer behavior towards reducing metal waste.
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Question #4
Find the measure of the indicated arc.
160 °
D
R
?
U
S
56°
T
Answer:
D. 48
Step-by-step explanation:
When a tangent and a secant, two secants, or two tangents intersect outside a circle then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.
56 = 1/2(160 - ?)
112 = 160 - ?
? = 160 - 112 = 48
Question #1
Solve for x
E
16x9
D
C
45°
Answer:
[tex]x = \frac{3\pi}{64} +\frac{9}{16}[/tex]
Step-by-step explanation:
We have
∠D = 45° = 45* π/180 radians = π/4 radians - eq(1)
big arc + small arc = 2π
small arc = 16x - 9
⇒ big arc = 2π - small arc
big arc = 2π - 16x + 9
[tex]\angle D = \frac{big \;arc - small \;arc}{2}[/tex]
[tex]\angle D = \frac{2\pi - 16x + 9 - 16x +9}{2}\\\\= \angle D = \frac{2\pi - 32x + 19 }{2}\\\\\angle D = \pi - 16x + 9[/tex]
Equating with eq(1)
π - 16x + 9 = π/4
⇒ 16 x = π - (π/4) +9
⇒ 16 x = (3π/4) +9
⇒ [tex]x = \frac{1}{16} (\frac{3\pi}{4} +9)[/tex]
[tex]x = \frac{3\pi}{64} +\frac{9}{16}[/tex]
A = –5(6t – 7) + 11. B = 3(x – 5) – 3(x + 5).
B = 8 + 2y – 5(2y – 6) + 4.
C = –5z + 5z(z – 3) – 7(6 – 8z).
Answer: the answer is 110.8
Step-by-step explanation: add um all up
Given the functions, f(x) = x2 + 2 and g(x) = 4x - 1, perform the indicated operation. When applicable, state the domain restriction.
The indicated operation is the composition of functions. To perform this operation, we substitute the expression for g(x) into f(x). The composition of f(g(x)) is given by f(g(x)) = (4x - 1)^2 + 2.
To compute f(g(x)), we first evaluate g(x) by substituting x into the expression for g(x): g(x) = 4x - 1. Next, we substitute this result into f(x): f(g(x)) = f(4x - 1).
Now, let's expand and simplify f(g(x)):
f(g(x)) = (4x - 1)^2 + 2
= (4x - 1)(4x - 1) + 2
= 16x^2 - 8x + 1 + 2
= 16x^2 - 8x + 3.
The domain of f(g(x)) is the same as the domain of g(x) since the composition involves g(x). In this case, g(x) is defined for all real numbers. Therefore, the domain of f(g(x)) is also all real numbers.
In summary, the composition of f(g(x)) is given by f(g(x)) = 16x^2 - 8x + 3, and the domain of f(g(x)) is all real numbers.
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19
Select the correct answer.
This table represents function f.
0
2
I
f(x)
0
-2
If function g is a quadratic function that contains the points (-3, 5) and (0, 14), which statement is true over the inter
-3
-4.5
-2
-2
-1
-0.5
1
-0.5
3
-4.5
OA. The average rate of change of fis less than the average rate of change of g.
O B.
The average rate of change of fis more than the average rate of change of g.
'O C.
The average rate of change of fis the same as the average rate of change of g.
OD. The average rates of change of f and g cannot be determined from the given information.
The correct statement is OB. The average rate of change of f is more than the average rate of change of g.
To determine the average rate of change (slope) of the functions f and g, we can use the formula:
Average Rate of Change = (f(x2) - f(x1)) / (x2 - x1)
For function f, using the given table, we can calculate the average rate of change between the points (0, 0) and (2, -2):
Average Rate of Change (f) = (-2 - 0) / (2 - 0) = -2 / 2 = -1
For function g, using the given points (-3, 5) and (0, 14), we can calculate the average rate of change:
Average Rate of Change (g) = (14 - 5) / (0 - (-3)) = 9 / 3 = 3
Comparing the average rates of change, we find that the average rate of change of f is -1, while the average rate of change of g is 3.
Therefore, the correct statement is:
OB. The average rate of change of f is more than the average rate of change of g.
The average rate of change of f is greater than the average rate of change of g, indicating that the function f is increasing at a faster rate than function g over the given interval.
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Which statement can be concluded using the true statements shown?
If two angles in a triangle measure 90° and x degrees, then the third angle measures (90-x) degrees.
In triangle ABC, angle A measures 90 degrees and angle B measures 50°.
Angle C must measure 50 degrees.
Angle C must measure 40 degrees.
O Angle C must measure (90 - 40) degrees.
O Angle C must measure (90-30) degrees.
Answer:
Angle C must measure 40 degrees.
Step-by-step explanation:
All angles in a triangle add up to 180 degrees
(90-50)=40 degrees
We can check our answer by adding all the angles up
90+50+40=180
Angle C must be 40 degrees
what is the equation of this line?
The calculated equation of the line is y = -2x + 3
How to calculate the equation of the lineFrom the question, we have the following parameters that can be used in our computation:
The graph
Where, we have
(1, 1) and (0, 3)
The equation of the line is calculated as
y = mx + c
Where
c = y when x = 0
So, we have
y = mx + 3
Using the points, we have
m + 3 = 1
So, we have
m = -2
This means that
y = -2x + 3
Hence, the equation of the line is y = -2x + 3
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at castleton university alex bought three mathematics textbook and four programming textbooks athe same school rick bought eight mathematic textbooks and a single programming textbook of alex spent 854.14 rick spend 1866.39 on textbooks what was the average cost of each book
Answer:
math = 227.98
programming = 42.55
Step-by-step explanation:
We have
3m + 4p = 854.14 -eq(1)
8m + 1p = 1866.39 -eq(2)
rq(2) x 4: 32m + 4p = 7465.56 -eq(3)
eq(3)-eq(1):
32m + 4p = 7465.56
- ( 3m + 4p = 854.14)
--------------------------------
29m = 6611.42
--------------------------------
⇒ m = 6611.42/29
m = 227.98
sub in eq(1)
3(227.98) + 4p = 854.14
4p = 854.14 - 683.94
4p = 170.2
p = 170.2/4
p = 42.55
Please answer ASAP I will brainlist
Answer:
(a) $556 billion
(b) $581 billion
(c) $693 billion
Step-by-step explanation:
The given function is:
[tex]\boxed{A(x)=314e^{0.044x}}[/tex]
where A(x) is the assets (in billions of dollars) for a financial firm .
If x = 7 corresponds to the year 2007 then:
x = 13 corresponds to the year 2013.x = 14 corresponds to the year 2014.x = 18 corresponds to the year 2018.Therefore, to find the assets for each of the given years, substitute the corresponding value of x into the function.
[tex]\begin{aligned}A(13)&=314e^{0.044 \cdot 13}\\&=314e^{0.572}\\&=314(1.77180712...)\\&=556.34743707...\\&=556\; \sf (nearest\;billion)\end{aligned}[/tex]
[tex]\begin{aligned}A(14)&=314e^{0.044 \cdot 14}\\&=314e^{0.616}\\&=314(1.851507181...)\\&=581.3732549...\\&=581\; \sf (nearest\;billion)\end{aligned}[/tex]
[tex]\begin{aligned}A(18)&=314e^{0.044 \cdot 18}\\&=314e^{0.792}\\&=314(2.20780762...\\&=693.2515954...\\&=693\; \sf (nearest\;billion)\end{aligned}[/tex]
Use the side lengths to prove which triangles form a right triangle.
Select all the triangles that form a right triangle
The side length that prove a right angle triangle is √2, √3 and √5.
How to find the side of a right triangle?A right angle triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.
Therefore, a right angle triangle can be proved by using the Pythagoras's theorem as follows:
Hence,
c² = a² + b²
where
c = hypotenuse sidea and b are the other legsTherefore,
(√2)² + (√3)² = (√5)²
Hence, the right angle triangle is the triangle with sides √2, √3 and √5.
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will give 100 points The box plots display measures from data collected when 15 athletes were asked how many miles they ran that day.
A box plot uses a number line from 0 to 13 with tick marks every one-half unit. The box extends from 1 to 3.5 on the number line. A line in the box is at 2. The lines outside the box end at 0 and 5. The graph is titled Group A's Miles, and the line is labeled Number of Miles.
A box plot uses a number line from 0 to 13 with tick marks every one-half unit. The box extends from 1 to 5 on the number line. A line in the box is at 2.5. The lines outside the box end at 0 and 11. The graph is titled Group C's Miles, and the line is labeled Number of Miles.
Which group of athletes ran the least miles based on the data displayed?
Group A, with a median value of 2 miles
Group C, with a median value of 2.5 miles
Group C, with a narrow spread in the data
Group A, with a wide spread in the data
Si 3,390 kg de plomo ocupan un volumen de 0.3m3. Encuentra la densidad del plomo
The density of lead is 11.3 kg/m³.
The density of lead can be calculated by using the formula D = M/V, where D represents density, M represents mass and V represents volume. The density of lead is the ratio of the mass of lead to the volume occupied by it.
Density of Lead:
Given that the lead has a mass of 3.390 kg and occupies a volume of 0.3 m³.
Density of Lead (D) = Mass of Lead (M) / Volume of Lead (V)D = 3.390 kg / 0.3 m³D = 11.3 kg/m³
Therefore, the density of lead is 11.3 kg/m³.
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Let A be the point (7,4) and D be (-5, -3). What is the length of the shortest path ABCD, where B is a point (x, 2) and C is a point (x,0)? This path consists of three connected segments, with the middle one vertical.
The length of the shortest path ABCD is 7 units.
To find the length of the shortest path ABCD, we need to determine the coordinates of points B and C and then calculate the distances between these points.
Given that B has a y-coordinate of 2, it lies on a horizontal line. Therefore, the y-coordinate of B is 2, and the x-coordinate is the same as the x-coordinate of point A, which is 7. So, B is the point (7, 2).
Similarly, C lies on a vertical line, and its x-coordinate is the same as the x-coordinate of point D, which is -5. So, C is the point (-5, 0).
Now, we can calculate the distances between the points. The distance between A and B can be found using the distance formula:
AB = √[tex]((x2 - x1)^2 + (y2 - y1)^2[/tex])
Substituting the coordinates of A and B, we have:
AB = √[tex]((7 - 7)^2 + (2 - 4)^2) = √(0^2 + (-2)^2[/tex]) = √4 = 2
The distance between B and C is simply the difference in their y-coordinates:
BC = |y2 - y1| = |2 - 0| = 2
Finally, the distance between C and D can be calculated using the distance formula:
CD = √[tex]((-5 - (-5))^2 + (0 - (-3))^2)[/tex] = √[tex](0^2 + 3^2)[/tex] = √9 = 3
Therefore, the length of the shortest path ABCD is the sum of the distances AB, BC, and CD:
Shortest path ABCD = AB + BC + CD = 2 + 2 + 3 = 7
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There are 6 horses in a race. How many ways can the first three positions of the order of the finish occur assume there are no ties
A square on a coordinate plane is translated 9 units down and 1 unit to the right. Which function rule describes the translation?
T1, –9(x, y)
T–1, –9(x, y)
T–9, 1(x, y)
T–9, –1(x, y)
The function rule that describes the given translation is T-9, 1(x, y).
The first value in the function rule represents the horizontal translation, while the second value represents the vertical translation. In this case, the square is translated 1 unit to the right, indicating a positive horizontal translation.
Additionally, the square is translated 9 units down, indicating a negative vertical translation. Therefore, the correct function rule is T-9, 1(x, y).
In the coordinate plane, the x-coordinate represents the horizontal position, and the y-coordinate represents the vertical position. When we apply the function rule T-9, 1 to the coordinates of the square, we subtract 9 from the y-coordinate and add 1 to the x-coordinate.
This results in the square being moved 9 units down and 1 unit to the right from its original position.
The negative sign in front of the 9 indicates a downward movement, and the positive sign in front of the 1 indicates a rightward movement. Hence, the translation is accurately described by the function rule T-9, 1(x, y).
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Answer:
C
Step-by-step explanation:
What is the equation in point slope form of the line that is perpendicular to the given line and passes through the point(2,5)?
Answer:
Step-by-step explanation:
To find the equation of a line that is perpendicular to a given line and passes through a specific point, we need to follow a few steps:
Find the slope of the provided line.
The point-slope form of a line is given by: y - y1 = m(x - x1), where (x1, y1) represents the given point.
Substituting the values, the equation of the perpendicular line becomes:
y - 5 = (-1/m)(x - 2)
Simplifying the equation further, we can rewrite it in point-slope form:
y - 5 = (-1/m)x + (2/m)
Find the indefinite integral. (Use C for the constant of integration.)
1. v + 1/
(2v − 20)^5dv
2. x^2/
x − 5 dx
3. x cos 8x2 dx
4. 176/e^−x + 1 dx
5.
1. The indefinite integral of (v + 1) / (2v - 20)^5 dv is -1 / (8(2v - 20)^4) + C.
2. The indefinite integral of x^2 / (x - 5) dx is (1/2) x^2 + 5x + 25 ln|x - 5| + C.
3. The indefinite integral of x cos(8x^2) dx is (1/16) sin(8x^2) + C.
4. The indefinite integral of 176 / e^(-x) + 1 dx is 176 ln|1 + e^x| + C.
1. To find the indefinite integral of (v + 1) / (2v - 20)^5 dv:
Let u = 2v - 20. Then du = 2 dv.
The integral becomes:
(1/2) ∫ (1/u^5) du
Now we can integrate using the power rule:
(1/2) ∫ u^(-5) du
Applying the power rule, we get:
(1/2) * (u^(-4) / -4) + C
= -1 / (8u^4) + C
Substituting back u = 2v - 20:
= -1 / (8(2v - 20)^4) + C
Therefore, the indefinite integral of (v + 1) / (2v - 20)^5 dv is -1 / (8(2v - 20)^4) + C.
2. To find the indefinite integral of x^2 / (x - 5) dx:
We can use polynomial long division to simplify the integrand:
x^2 / (x - 5) = x + 5 + 25 / (x - 5)
Now we can integrate each term separately:
∫ x dx + ∫ (5 dx) + ∫ (25 / (x - 5) dx)
Using the power rule, we get:
(1/2) x^2 + 5x + 25 ln|x - 5| + C
Therefore, the indefinite integral of x^2 / (x - 5) dx is (1/2) x^2 + 5x + 25 ln|x - 5| + C.
3. To find the indefinite integral of x cos(8x^2) dx:
We can use the substitution method. Let u = 8x^2, then du = 16x dx.
The integral becomes:
(1/16) ∫ cos(u) du
Integrating cos(u), we get:
(1/16) sin(u) + C
Substituting back u = 8x^2:
(1/16) sin(8x^2) + C
Therefore, the indefinite integral of x cos(8x^2) dx is (1/16) sin(8x^2) + C.
4. To find the indefinite integral of 176 / e^(-x) + 1 dx:
We can simplify the integrand by multiplying the numerator and denominator by e^x:
176 / e^(-x) + 1 = 176e^x / 1 + e^x
Now we can integrate:
∫ (176e^x / 1 + e^x) dx
Using u-substitution, let u = 1 + e^x, then du = e^x dx:
∫ (176 du / u)
Integrating 176/u, we get:
176 ln|u| + C
Substituting back u = 1 + e^x:
176 ln|1 + e^x| + C
Therefore, the indefinite integral of 176 / e^(-x) + 1 dx is 176 ln|1 + e^x| + C.
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