The distance between the points (4, 5) and (2, 2) is approximately 3.61 units.
What is the distance?
To find the distance between two points, we can use the distance formula:
distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
Using the given points (4, 5) and (2, 2), we have:
distance = √[(2 - 4)² + (2 - 5)²]
= √[(-2)² + (-3)²]
= √[4 + 9]
= √13
So the distance between the points (4, 5) and (2, 2) is approximately 3.61 units.
To graph these points, we would plot them on a coordinate plane. The point (4, 5) would be two units to the right of the y-axis and five units up from the x-axis.
The point (2, 2) would be two units to the right of the y-axis and two units up from the x-axis. We could then use a ruler to measure the distance between the two points on the graph to verify that it is approximately 3.61 units.
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The data shown on the scatter plot below demonstrates the relationship between the time of day and the total number of calories a teenager consumes throughout the day.
The slope of the best-fit line shows that as time
, the total number of calories that a teenager consumes throughout the day
.
Answer:
The slope of the best-fit line shows that as time increases, the total number of calories that a teenager consumes throughout the day also increases.
For Assessment 2 you are required to conduct a research project where you collect Data, Analysis and Present.
For your Assessment 2 your Presentation must include -
1. What your study of data is on
2. A prediction of what you think the data will reveal.
3. How you collected your data
4. A table of figures – the data you have collected
5. The medium, mean and mode of your data.
6. A graph of your data – you choose the type (pie/line). Preferably created digitally.
7. A written analysis of your graph using graph language
8. What you found in your study; how close you were with your prediction?
PLS HELP!!
The findings of the research project suggest that regular exercise can be an effective tool for managing stress in university students.
What would be the research project?A sample research project is given below:
Study of Data: The impact of exercise on stress levels in university students
Prediction: We predict that there will be a significant decrease in stress levels following a four-week exercise program.
Data Collection: Participants were recruited through flyers and social media. They were asked to complete a pre-test stress survey before beginning the exercise program, and a post-test stress survey after completing the program. The exercise program consisted of a combination of cardiovascular and resistance training, meeting three times a week for four weeks.
Table of Figures:
Participant Pre-Test Stress Score Post-Test Stress Score
1 32 20
2 28 17
3 36 25
... ... ...
30 41 22
Measures of Central Tendency:
Median Pre-Test Stress Score: 34Median Post-Test Stress Score: 21Mean Pre-Test Stress Score: 33.7Mean Post-Test Stress Score: 22.5Mode Pre-Test Stress Score: 32Mode Post-Test Stress Score: 22Graph: Line graph showing the change in stress levels over time, with pre-test scores in blue and post-test scores in red.
Written Analysis: The line graph shows a consistent decrease in stress levels across all participants following the four-week exercise program. The majority of participants showed a decrease in stress levels of 10 or more points, indicating a significant reduction in stress. There is a clear pattern of improvement in stress levels over time, with the post-test scores consistently lower than the pre-test scores.
Findings: Our study found a significant decrease in stress levels following a four-week exercise program in university students. Our prediction was accurate, as we anticipated a decrease in stress levels.
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which of the following is a graph of x^2>25
20 POINTSSSSSSS
Plotting the graph of the inequality we get a line passing through the point x = 5.
What is inequality?A mathematical notion called inequality examines two numbers or expressions to see if they are equal or not. When there is an inequality, one value is either more than or less than the other. "" (less than), ">" (greater than), "" (less than or equal to), and "" are used to denote inequality (greater than or equal to). In algebra and calculus, inequalities are frequently used to explain how variables relate to one another and to solve equations and systems of equations. In the actual world, inequalities are frequently used to compare measurements, quantities, and other variables.
The given function is x² > 25.
Simplifying the given inequality we have:
x > √25
x > 5 or -5
But - 5 is < x
So we have,
-5 < x < 5
Plotting the graph of the inequality we get a line passing through the point -5 < x < 5.
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Twenty randomly selected students took the Biology final examination. If the sample mean was 92 and the standard deviation was 9.38.
What is the margin of error using 99% confidence interval?
A
The margin of error using a 99% confidence interval is 6.21. This means that we can be 99% confident that the true mean score of all Biology students (not just the sample) falls within a range of 92 ± 6.21.
What is percentage?
Percentage is a way of expressing a number or proportion as a fraction of 100. It is represented by the symbol "%".
To calculate the margin of error using a 99% confidence interval, we need to use the formula:
Margin of error = z* (standard deviation / √n)
Where:
z* = the z-score associated with the confidence level, which is 2.576 for a 99% confidence level (using a standard normal distribution table).
standard deviation = 9.38
n = 20
Plugging in the values, we get:
Margin of error = 2.576 * (9.38 / √20)
= 6.21 (rounded to two decimal places)
Therefore, the margin of error using a 99% confidence interval is 6.21. This means that we can be 99% confident that the true mean score of all Biology students (not just the sample) falls within a range of 92 ± 6.21.
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Find the measure of an angle with measure between 0° and 360° that is coterminal with an angle measuring â€""800°. °
The angle measuring 280° is coterminal with the angle measuring -800° and lies between 0° and 360°.
To find the coterminal angle between 0° and 360° for an angle measuring -800°, follow these steps:
1. Divide the given angle (-800°) by 360° to determine how many full rotations are made:
-800° ÷ 360° = -2.22.
2. Since we are looking for a positive coterminal angle, round the result down to the nearest whole number:
-2.22 rounds down to -3.
This tells us there are three full negative rotations.
3. Multiply the whole number (-3) by 360° to find the total angle of the rotations:
-3 × 360° = -1080°.
4. Add the total angle of the rotations to the given angle to find the coterminal angle between 0° and 360°: -800° + 1080° = 280°.
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Solve the following equations. Show your solution.
1.) 1/5x - 2 = -3
2.) 15 = 10 - x/4
3.) x/3 - 12 = -2
4.) x/-2 + (-6) = 6
Answer:
Step-by-step explanation:
1.) 1/5x - 2 = -3
Adding 2 to both sides, we get:
1/5x = -1
Multiplying both sides by 5, we get:
x = -5
Therefore, the solution is x = -5.
2.) 15 = 10 - x/4
Subtracting 10 from both sides, we get:
5 = -x/4
Multiplying both sides by -4, we get:
-20 = x
Therefore, the solution is x = -20.
3.) x/3 - 12 = -2
Adding 12 to both sides, we get:
x/3 = 10
Multiplying both sides by 3, we get:
x = 30
Therefore, the solution is x = 30.
4.) x/-2 + (-6) = 6
Adding 6 to both sides, we get:
x/-2 = 12
Multiplying both sides by -2, we get:
x = -24
Therefore, the solution is x = -24.
Write a polynomial f(x) that satisfies the given conditions.
Polynomial of lowest degree with zeros of -4(multiplicity 1) 1( multiplicity 2)and with F(0)=-12
If -4 is a zero with multiplicity 1, then (x + 4) is a factor of the polynomial. Similarly, if 1 is a zero with multiplicity 2, then (x - 1)^2 is a factor of the polynomial. Therefore, we can write the polynomial in factored form as:
f(x) = a(x + 4)(x - 1)^2
where "a" is a constant that we need to determine.
To find "a", we use the fact that f(0) = -12. Substituting x = 0 into the equation above, we get:
f(0) = a(0 + 4)(0 - 1)^2
-12 = -4a
Solving for "a", we get:
a = 3
Therefore, the polynomial is:
f(x) = 3(x + 4)(x - 1)^2
Note that this polynomial has a zero at x = -4 (with multiplicity 1), a zero at x = 1 (with multiplicity 2), and f(0) = -12.
Solve the inequality and graph the solution.
8>
–
3a+2
To draw a ray, plot an endpoint and select an arrow. Select an endpoint to change it from closed to open. Select the middle of the ray to delete it.
The solution is greater than [tex]-2[/tex]
Define inequalityA mathematical assertion known as an inequality contrasts two values or expressions, demonstrating their relationship in terms of greater than[tex]\geq[/tex], less than (), greater than or equal to [tex]\geq[/tex], or less than or equal to ().
When two quantities are related in a manner other than being equal, it is referred to as an inequality.
[tex]8 > -3a+2[/tex]
Subtracting [tex]2[/tex] from both sides we get:
[tex]6 > -3a[/tex]
Dividing both sides by [tex]-3[/tex] and reverse the inequality sign when dividing by negative number
[tex]a > -2[/tex]
Since a cannot equal [tex]-2[/tex], we can plot an open circle at this value to represent the solution on a number line. We can also depict an arrow pointing up the number line to represent that an is larger than [tex]-2[/tex]
The arrow denotes any value higher than -2, and the open circle at -2 denotes that -2 is not part of the solution.
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Answer:
a>-2
explained:
-3a+2<8
−3+2−2<8−2
−3a<6
−[tex]\frac{3a}{3}[/tex]<[tex]\frac{6}{-3}[/tex]
a>−2
2[tex]x^{2} =1[/tex]
Answer:
Step-by-step explanation:
x = [tex]+-\sqrt[]{\frac{1}{2} }[/tex]
Verify that the following identity is true. You must show all work to receive credit! (1 - cos a) (1 + cot? a) = 1
Answer:
To verify the given identity:
(1 - cos a) (1 + cot a)
= (1 - cos a) (1 + cos a / sin a) [since cot a = cos a / sin a]
= 1 - cos^2 a / sin a + cos a - cos^2 a / sin a
= 1 - (cos^2 a + cos^2 a) / sin a + cos a
= 1 - 2 cos^2 a / sin a + cos a
= 1 - 2 (1 - sin^2 a) / sin a + cos a [since cos^2 a = 1 - sin^2 a]
= 1 - 2 / sin a + 2 sin a / sin a + cos a
= 1 - 2 / sin a + 2 + cos a
= 1 + 2 (1 - sin a) / sin a
= 1 + 2 cos^2 a / sin a
= 1 + 2 cot^2 a
= (1 + cot^2 a) + 2 cot^2 a
= cosec^2 a + 2 cot^2 a
= 1 + cot^2 a [since cosec^2 a = 1 + cot^2 a]
Therefore, (1 - cos a) (1 + cot a) = 1 is true.
Find the value of X tell whether the side links from a pen Taghreed triple number 42
The value of x is given as 9.8
How to solve for xWe have to solve for X using the Pythagorean theorem
The Pythagorean theorem is a mathematical formula that relates to the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In other words:
c² = a² + b²
Where:
c is the length of the hypotenuse
a and b are the lengths of the other two sides
14² = x² + 10²
196 = x² + 100
196 - 100 = x²
x = √96
x = 9.8
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a survey asks a random sample of 1500 adults in ohio if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to education. let ^ p denote the proportion in the sample who say they support the increase. suppose that 8% of all adults in ohio support the increase. the standard deviation of the sampling distribution is
The standard deviation of the sampling distribution is 0.00702.
In this case, we know that 8% of all adults in Ohio support the increase. We can use this information, along with the sample size of 1500, to calculate the standard deviation of the sampling distribution. The formula for the standard deviation of the sampling distribution is:
standard deviation = √ [(population proportion x (1 - population proportion)) / sample size]
Plugging in the numbers, we get:
standard deviation = √ [(0.08 x 0.92) / 1500]
standard deviation = √0.00004928
standard deviation = 0.00702
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What is the prime? what is P for?
The probability of prime number in a die would be ½.
If a die rolls, the total outcome would be 6.
Number of possible outcomes = 3( as there are only 3 prime numbers in the die that are 2,3 and 5).
P(prime number) = 3/6
So the probability of the prime number in the die would be ½.
Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will learn the potential outcomes for a random experiment using this fundamental theorem of probability, which is also applied to the probability distribution. Knowing the total number of outcomes is necessary before we can calculate the likelihood that a specific event will occur.
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Irma measured the floor of her storage unit, which is rectangular. It is 5 feet wide and 13 feet from one corner to the opposite corner. How long is the storage unit?
According to given information, the length of the rectangular storage unit is 12 feet.
What is rectangle?
A rectangle is a two-dimensional shape with four sides and four right angles. It is a quadrilateral with opposite sides that are parallel and equal in length.
We can use the Pythagorean theorem to find the length of the storage unit. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the 5-foot-wide floor and the length of the storage unit form the two sides of a right triangle, with the diagonal (the distance from one corner to the opposite corner) being the hypotenuse. We can set up the equation as:
[tex]hypotenuse^2 = side1^2 + side2^2[/tex]
where side1 = 5 feet and hypotenuse = 13 feet.
Simplifying this equation, we get:
[tex]hypotenuse^2[/tex][tex]= 5^2 + side2^2[/tex]
169 = 25 + [tex]side2^2[/tex]
[tex]side2^2[/tex] = 144
side2 = 12
Therefore, the length of the storage unit is 12 feet.
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The value of a brand new car is $27,000 and the value depreciates 23% every year. Write a function to represent the value of the car after t years, where the monthly rate of change can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage rate of change per month, to the nearest hundredth of a percent.
[tex]27000(0.77^t)[/tex]and the percentage rate of change per month is 3.13%.
What is the percentage rate?
The term annual percentage rate of charge refers to the interest rate for an entire year rather than just a monthly fee or rate as applied on a loan, home loan, credit card, etc. It can also be referred to as a nominal APR or an effective APR. It is an annual rate of a finance charge.
Here, we have
The value of a brand-new car is $27,000 and the value depreciates 23% every year.
we have to write a function to represent the value of the car after t years.
The coefficient of the function is 0.77
To find the rate of change per month, we need to find the rate at which the value of the car is decreasing each month.
we can use the rule of 72
72/r = t where r is the rate of change per month
r = 72/t
[tex]r = 72/23 = 3.13[/tex] (approx)
So the percentage rate of change per month is 3.13%
Hence, The function to represent the value of the car after t years is V(t) = [tex]27000(0.77^t)[/tex] and the percentage rate of change per month is 3.13%.
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Find the value of x y and z
The values of the variables x = 5√5, z = 20 and y = 10√5.
What is triangle?A triangle is a geοmetric shape with three sides and three angles. It is a pοlygοn and οne οf the simplest and mοst cοmmοn shapes in geοmetry. The three sides οf a triangle can be οf different lengths οr the same length, and the three angles can alsο vary in size. The sum οf the angles in a triangle is always 180 degrees. Triangles are used in many applicatiοns in mathematics, science, engineering, and everyday life.
Here,
To solve this problem, we can use the Pythagoras theorem, which is:
c² = a² + b²
In the smallest triangle:
x² = 5² + 10²
x² = 125
x = √125
x = 5√5
Then for medium triangle:
y² = 10² + z²
Again for Largest triangle:
(5 + z)² = (5√5)² + y²
y² = (5 + z)² - (5√5)²
Both formula have like term y², So taking it common we have:
(5 + z)² - (5√5)² = 10² + z²
Solve for Z :
(5 + z)² - 125 = 100 + z²
+ z² + 10z - 125 = 100 + z²
z² - z² + 10z - 125 + 25 = 100
10z = 100 + 125 - 25
10z = 200
z = 200/10
z = 20
When z = 20. Then y will be
y² = 10² + z²
y² = 10² + 20²
y² = 100 + 400
y² = 500
y = √500
y = √500
y = 10√5
Thus, We have the values of the variables x = 5√5, z = 20 and y = 10√5.
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Select all expressions that are equivalent to
0.75x + 0.25(x + 12.4) + (x – 2.1).
a.
x + 3.1 + x + 2.1
b.
x + 3.1 + x – 2.1
c.
2x + 1
d.
x + 1
Answer:
B and C are the answers according to the expression
In both places where you can place point C to form the right triangle CDL times d times l, the horizontal distance between points Dd and Ll, |x2−x1|= (blank) blocks, and the vertical distance between points Dd and Ll, |y2−y1|= (blank) blocks.
Fill the blank and answer
In a right triangle CDL, the horizontal distance between points D and L is |x2 - x1| blocks, and the vertical distance between points D and L is |y2 - y1| blocks.
We need to determine the horizontal and vertical distances between points D (x1, y1) and L (x2, y2) in a right triangle CDL.
Identify the coordinates of points D and L.
Let D = (x1, y1) and L = (x2, y2)
Calculate the horizontal distance between points D and L.
Horizontal distance = |x2 - x1|
Calculate the vertical distance between points D and L.
Vertical distance = |y2 - y1|
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Psychology. Which of the following has NOT been found to impair immune functioning?
a. Loneliness.
b. Marital problems. X
C. Seeing oneself as low on the social ladder.
d. All of these impair immune functioning.
Answer:
The answer is b. Marital problems.
Step-by-step explanation:
Marital problems have not been consistently found to impair immune functioning.
ty suspects are properly judged while, of course, 10% of the guilty suspects are improperly found innocent. on the other hand, innocent suspects are misjudged 1% of the time. if the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocen
The probability is approximately 17.43%.
How to find probability?Use Bayes' theorem. Bayes' theorem states that is:
P(A|B) = (P(B|A) * P(A)) / P(B),
P(A): Probability of being innocent = 95% (since 5% have committed a crime)P(A'): Probability of being guilty = 5%P(B|A): Probability that the serum indicates guilt given that the suspect is innocent = 1%P(B|A'): Probability that the serum indicates guilt given that the suspect is guilty = 90% (since 10% are improperly found innocent)Using the law of total probability: P(B) = P(B|A) * P(A) + P(B|A') * P(A')
P(B) = (0.01 * 0.95) + (0.9 * 0.05) = 0.0095 + 0.045 = 0.0545
Apply Bayes' theorem to find P(A|B):
P(A|B) = (P(B|A) * P(A)) / P(B) = (0.01 * 0.95) / 0.0545 ≈ 0.1743
So, the probability that the suspect is innocent given that the serum indicates guilt is approximately 17.43%.
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please helppp A.S.A.P! What is the surface area of this right triangular prism?
Enter your answer in the box.
The surface area of the given right triangular prism is 84 in².
How to find the surface area of a right triangular prism?
To calculate the surface area of a right triangular prism, we need to find the area of all the faces and add them up.
First, let's find the area of the triangular bases. The base of the triangle is 8 in, and the height is 3 in, so the area of each triangular base is [tex]\frac{1}{2} \times base \times height = \frac{1}{2} \times 8 \times 3 = 12 \: {in}^{2}[/tex]
There are two triangular bases, so the total area of both bases is (2 × 12) in²= 24 in²
Now, let's find the area of the rectangular faces. The length of the rectangle is 5 in, and the breadth is 4 in.
So, the area of each rectangular face is length × breadth = 5 in × 4 in = 20 in²
There are three rectangular faces, so the total area of all three faces is 3 × 20 in² = 60 in².
Finally, we add the areas of the bases and the rectangular faces to get the total surface area.
Total surface area = 24 in² + 60 in² = 84 in²
Therefore, the surface area of this right triangular prism is 84 square inches.
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Label the measures of all of the angles on the picture
below.
Remember that, if you know one angle is 64°, you can use
this to help you figure out the measure of all of the other
angles.
→ Do NOT use a protractor. Use what you know about angle
relationships.
Answer:
Step-by-step explanation:
its doc sus
which of the following statements explains why nonprobability sampling carries more risk of selection bias than probability sampling? group of answer choices exclusion criteria are not used in nonprobability sampling nonprobability sampling does not use randomization nonprobability sampling uses strata that are not mutually exclusive exclusion criteria limit the representativeness of the sample
Nonprobability sampling carries more risk of selection bias than probability sampling because nonprobability sampling does not use randomization.What is nonprobability sampling?A nonprobability sampling is a method of selecting participants for a study that does not allow the researcher to use a random selection process.
Nonprobability sampling, also known as purposive sampling, is a technique that involves choosing participants based on subjective criteria, such as availability or willingness to participate. This method is frequently used in research that investigates hard-to-reach populations, such as homeless individuals or drug addicts.
A key feature of probability sampling is that it allows the researcher to obtain a representative sample of the population, minimizing the risk of selection bias. In contrast, nonprobability sampling does not provide the same level of assurance that the sample will be representative. Because participants are chosen based on subjective criteria rather than random selection, there is a greater risk of selection bias.
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true or false the polynomial 2y^2 -15y-8 is prime
Answer:
The polynomial 2y^2 -15y-8 is not prime. It can be factored as (2y+1)(y-8).
Answer:false
Step-by-step explanation:
. solve the problem as an lp, ignoring the integer constraints. b. what solution is obtained by rounding up fractions greater than or equal to 1/2? is this the optimal integer solution? c. what solution is obtained by rounding down all fractions? is this the optimal integer solution? explain. d. show that the optimal objective function value for the ilp (integer linear programming) is lower than that for the optimal lp.
As per the information provided, the answer to all the parts in the question will be as follows:
a. To solve the LP problem by ignoring the integer constraints in Excel using Solver, we can follow the steps below:
Enter the objective function and the constraints in a new Excel worksheet: Objective function: Maximize 15x1 + 2x2 Constraints: 7x1 + x2 <= 23 3x1 - x2 <= 5
Open the Solver add-in by clicking on Data -> Solver in the Excel menu.
Set the objective function to maximize and set the variable cells to x1 and x2. Set the constraints by clicking on Add in the Solver Parameters dialog box.
Set the Solver options to "Assume Linear Model" and "Make Unconstrained Variables Non-Negative". Click Solve. The solution to the LP problem is x1=3, x2=2.714, with an optimal objective function value of 51.714.
b. If we round up fractions greater than or equal to 1/2, the solution becomes x1=3, x2=3, with an objective function value of 51. This is not the optimal integer solution, as we will see in part d.
c. If we round down all fractions, the solution becomes x1=2, x2=2, with an objective function value of 34. This is not the optimal integer solution either, as we will see in part d.
d. To solve the ILP problem in Excel using Solver, we can follow the steps below:
Open the Solver add-in by clicking on Data -> Solver in the Excel menu. Set the objective function to maximize and set the variable cells to x1 and x2. Set the constraints by clicking on Add in the Solver Parameters dialog box. Set the Solver options to "Assume Linear Model" and "Make Unconstrained Variables Non-Negative". Add integer constraints by clicking on Add in the Solver Parameters dialog box, and setting the integer constraints for x1 and x2. Click Solve.
The solution to the LP problem is x1=2, x2=3, with an optimal objective function value of 48.
As we can see, the optimal objective function value for the LP problem (48) is lower than that for the LP problem (51.714), regardless of rounding up or down.
e. The optimal objective function value for the ILP problem is always less than or equal to the corresponding LP's optimal objective function value because the LP problem allows fractional solutions, while the ILP problem only allows integer solutions. Introducing additional constraints that restrict the variables to integers can only reduce the feasible solution space, and thus lead to a lower optimal objective function value. The LP and ILP problems would be equal if the optimal solution for the LP problem happens to be an integer solution.
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Note that the full question is:
Given the following all-integer linear program: (COMPLETE YOUR SOLUTION IN EXCEL USING SOLVER AND UPLOAD YOUR FILE. BE SURE THAT EACH WORKSHEET IN THE EXCEL FILE CORRESPONDS TO EACH QUESTION BELOW ) Max 15x1 + 2x2 s. t. 7x1 + x2 < 23 3x1 - x2 < 5 x1, x2 > 0 and integer a. Solve the problem (using SOLVER) as an LP, ignoring the integer constraints.
b. What solution is obtained by rounding up fractions greater than or equal to 1/2? Is this the optimal integer solution? c. What solution is obtained by rounding down all fractions? Is this the optimal integer solution? Explain. d. Show that the optimal objective function value for the ILP is lower than that for the optimal LP (Eg. Resolve original problem using SOLVER with the Integer requirement). e. Why is the optimal objective function value for the ILP problem always less than or equal to the corresponding LP's optimal objective function value? When would they be equal?
The actual German Grand Prix is 64 laps on the same track. On the second practice day, Valterri’s team wants him to race over 30 laps to make sure they are prepared. They record his times and laps in a table, shown below. What is Valterri’s unit rate for Day 2, in minutes per lap?
Given f(x) = x ^ 2 + 1 and f(g(x)) = 4x ^ 2 + 4x + 2 find g(x)
PLS HELP FAST!
Answer:
g(x) is the square root of 4x^2 + 4x + 1.
Step-by-step explanation:
We are given that f(x) = x^2 + 1 and f(g(x)) = 4x^2 + 4x + 2.
To find g(x), we need to substitute g(x) into the expression for f and simplify:
f(g(x)) = (g(x))^2 + 1 = 4x^2 + 4x + 2
Subtracting 1 from both sides, we get:
(g(x))^2 = 4x^2 + 4x + 1
Taking the square root of both sides (remembering to include both the positive and negative roots), we get:
g(x) = ±√(4x^2 + 4x + 1)
However, we need to choose the sign of g(x) such that f(g(x)) matches the given expression of f(g(x)) = 4x^2 + 4x + 2.
Let's try using the positive root first:
g(x) = √(4x^2 + 4x + 1)
Then we can find f(g(x)):
f(g(x)) = (g(x))^2 + 1 = 4x^2 + 4x + 2
This matches the given expression, so we can conclude that:
g(x) = √(4x^2 + 4x + 1)
Therefore, g(x) is the square root of 4x^2 + 4x + 1.
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The value of a brand new car is $27,000 and the value depreciates 23% every year. Write a function to represent the value of the car after t years, where the monthly rate of change can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage rate of change per month, to the nearest hundredth of a percent.
Answer and Explanation:
The value of a brand new car,
P
=
$
28000
The value depreciates every year,
r
=
19
%
Time,
t
=
1
4
Number of compounding period (Compounded quarterly),
n
=
4
Write a function to represent the value of the car after
t
years:
A
=
P
(
1
−
r
n
)
n
×
t
∴
A
=
28000
(
1
−
19
%
4
)
4
t
Let us evaluate the value of the car after one quarter:
A
=
28000
(
1
−
19
%
4
)
4
t
=
28000
(
1
−
19
100
4
)
4
×
1
4
=
28000
(
1
−
0.19
4
)
4
×
1
4
=
28000
(
1
−
0.19
4
)
1
=
28000
(
1
−
0.19
4
)
=
28000
(
1
−
0.0475
)
=
28000
(
0.9525
)
∴
A
=
26670
Therefore, the value of the car after one quarter is
$
26670
.
To calculate the percentage rate of change per quarter:
Percentage rate
=
P
−
A
P
×
100
%
=
28000
−
26670
28000
×
100
%
=
1330
28000
×
100
%
=
70
⋅
19
70
⋅
400
×
100
%
=
19
400
×
100
%
=
0.0475
×
100
%
∴
Percentage rate of change per quarter
=
4.75
%
Hence, the percentage rate of change per quarter is
4.75
%
.
a solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. the total volume of the solid is 10 cubic centimeters. find the radius r (in cm) and height h (in cm) of the cylinder that produces the minimum surface area.
The radius r that produces the minimum surface area is 1.06 cm and the height h is 1.41 cm.
Let r be the radius of the cylinder and h be the height. You can find the equation for the volume of a solid by adding the volumes of the two hemispheres and the cylinder.
Volume = [tex](2/3)πr^3 + πr^2h[/tex] = 10 cubic centimeters
Now let's find the values of r and h that minimize the surface area of the solid. This area consists of three parts:
Curved surfaces of two hemispheres and sides of a cylinder. This can be expressed as:
surface area = [tex]2πr^2 + 2πrh[/tex]
To find the values of r and h that minimize this expression, we can use the Lagrangian multiplier method. Given that the volume of the solid is 10 cubic centimeters, we want to minimize the surface area. So it looks like this:
[tex]f(r, h) = 2πr^2 + 2πrh + λ[(2/3)πr^3 + πr^2h - 10][/tex]
Taking the partial derivatives for r, h, and λ and setting them to zero gives
[tex]4πr + 2πh = 2πr^2λ + (4/3)πr^3λ[/tex]
[tex]2πr = πr^2λ + πrhλ[/tex]
Solving these equations simultaneously gives:
h = 4r/3
λ = 2/(3r)
Substituting these values into the volume equation gives:
[tex]r^2h = 5/3[/tex]
Substituting h = 4r/3 from above, we get:
[tex]r^3 = 15/8pi[/tex]
Taking the cube root of both sides gives:
r=1.06cm
Substituting this value for r into the formula for h yields:
h=1.41cm
Therefore, the radius r that produces the minimum surface area is about 1.06 cm and the height h is about 1.41 cm.
learn more about the Lagrangian multiplier method
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Solve for x. Round your answer to the nearest tenth if necessary. Figures are not
necessarily drawn to scale.
R
61°
55
47
52°
P
67⁰
T
67⁰
X
52%
61°
U
S
44
Given the similar triangles, Note that x = 51.2
What is the explanation for the above response?Since both triangles are proportional,
64/60 = x/48
To solve for x in the equation:
64/60 = x/48
We can cross-multiply to get rid of the fractions:
64 * 48 = 60 * x
3072 = 60x
Finally, we isolate x by dividing both sides by 60:
x = 3072/60 = 51.2
Therefore, the solution is:
x = 51.2
Learn more about similar triangles at:
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