Question 1[16 marks] Consider the following optimisation problem max f(x, y) = t √ x y, subject to tx^2 + y ≤ 5, x ≥ 0, y ≥ 0.

a) Solve the problem for t = 1.

b) State and explain the content of the envelope theorem.

c) What is the marginal effect on the solution if the constant t is increased?

Answer:

4w=25

Step-by-step explanation:

Answer: \((x+2)^{2}+(y-4)^{2} =16\)

and

\((x-6)^{2} +(y+4)^{2} =20\)

Step-by-step explanation:

Equation for a Circle

\((x-h)^{2} +(y-k)^{2} =r^{2}\)

h= -2 k= 4 r= 4

\((x+2)^{2}+(y-4)^{2} =16\)

Part 2

Complete the Square

\(x^{2} +y^{2} -12x+8y+32=0\\(x^{2}-12x+-)+(y^{2}+8y)=-32\\(x^{2} -12x+36)=(y^{2} +8y+16)=-32+36+16\\\\(x-6)^{2} +(y+4)^{2} =20\)

Answer:

1) Choice B

(x-2)^2 + (y+4)^2 = 16

2) Choice A

(x-6)^2 + (y+4)^2 = 20

Step-by-step explanation:

Equation for circle is (x-h)^2 + (y-k)^2 = r^2

(h,k) is cords

1) (x-2)^2 + (y+4)^2 = 16

Cords = (-2, 4)

radius = \(\sqrt{16}\) = 4

2) x^2 + y^2 - 12x +8y +32 = 0

x^2 - 12x + y^2 +8y +32 = 0

Choice A

(x-6)^2 + (y+4)^2 = 20

Multiply it out

x^2 - 12x + 36 + y^2 +8y + 16 = 20

Combine same values

x^2 - 12x + y^2 +8y + 52 = 20

Bring it all to one side (subtract)

x^2 - 12x + y^2 +8y +32 = 0

Variables that indicate the distance a target is from the level achieved are called performance metrics or progress indicators.


Variables that indicate the distance a target is from the level achieved are called performance metrics or progress indicators. These variables help measure and track progress towards a goal by providing a quantitative or qualitative measure of how far the target is from the desired level of achievement.

Performance metrics can be represented in various forms, such as distance covered, percentage completed, points earned, or even subjective evaluations. For example, in a fitness program, a performance metric could be the number of miles run or the number of push-ups completed.

These metrics enable individuals or organizations to assess their progress and make necessary adjustments to reach their desired outcomes.

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The sale price of each pen if 10 pens are purchased at the rate of Rs 12.30 each and there is a total gain of Rs 24 is Rs 14.70

Total number of pens = 10

Cost of each pen = Rs 12.30

Total gain = Rs 24

Gain on each pen = Total gain / Total number of pens

= Rs 24 / 10

= Rs 2.4

The sale price of each pen = Cost of each pen + Gain on each pen

= Rs 12.30 + Rs 2.4

= Rs 14.70

Hence, each pen is sold at Rs 14.70

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The limit can be expressed as a definite integral of the function y = f(x) on the interval [a, b] as ∫[a,b] f(x) dx. The evaluation of the definite integral depends on the specific function f(x) and the interval [a, b]. Interpreting the integral as an area, it represents the accumulated area under the curve of the function between the limits of integration [a, b].

To express the given limit as a definite integral, we start by considering the function y = f(x) and the interval [a, b]. The limit of the function as x approaches a can be written as lim[x→a] f(x). By expressing this limit as a definite integral, we have ∫[a,b] f(x) dx.

The definite integral represents the area under the curve of the function y = f(x) on the interval [a, b]. The integral sign, ∫, represents the summation of infinitely many small areas. The function f(x) determines the height of each infinitesimal rectangle, and dx represents the width. By integrating f(x) with respect to x over the interval [a, b], we calculate the total area enclosed between the curve and the x-axis.

To evaluate the definite integral in part (a), we need to know the specific function f(x) and the interval [a, b]. By evaluating the integral, we find the numerical value that represents the area under the curve. The evaluation of the definite integral can be done using various integration techniques, such as the fundamental theorem of calculus or integration rules specific to the function f(x)

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(OCF): OCF = Net Income + Depreciation & Amortization - Changes in Working Capital, (CapEx): CapEx = Investment in Property, Plant & Equipment + Investment in Intangible Assets. therefore, (FCFF): FCFF = OCF - CapEx

The Free Cash Flow to Firm (FCFF) is a measure of a company’s cash flow available for distribution to shareholders and debt holders. To calculate FCFF, the following three steps are taken:

1. Calculate Operating Cash Flow (OCF): OCF is calculated by taking net income, adding back depreciation and amortization, and subtracting any changes in working capital.

2. Calculate Capital Expenditures (CapEx): CapEx includes investments in property, plant, and equipment, as well as any investments in intangible assets.

3. Calculate Free Cash Flow to Firm (FCFF): FCFF is calculated by subtracting CapEx from OCF. This figure represents the cash flow available for distribution to shareholders and debt holders.

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Answer:

s = 11

Step-by-step explanation:

Given

\(\frac{r}{s}\) = 22 ← substitute r = 242

\(\frac{242}{s}\) = 22 ( multiply both sides by s )

242 = 22s ( divide both sides by 22 )

11 = s

Answer:

S would equal 11

Step-by-step explanation:

Answer:

Rotation because you can see that the shape is rotated on the other half of the graph.

Step-by-step explanation:

Answer:

a) y=11x+24

b) The slope of this equation is 11. You get the slope by taking the difference of the number of participants over the difference of the years: 35-57/1-3

c) The y intercept is 24 and it would be before the first year. (?)

d) There should be 79 participants in the fifth year.

Step-by-step explanation:

I hope this helped. I'm not 100 percent sure the explanation for letter c.

ANSWER:

B

Step-by-step explanation:

The rate at which water being poured into the cup when the water level is 4 cm is \(192\pi\) \(cm^3\)/sec

What is rate of change?

Suppose there is a function and there are two quantities. If one quantity of a function changes, the rate at which other quantity of the function changes is called rate of change of a function.

Here,

Rate of change of volume is being calculated

Radius of conical cup (r) = 20 cm

Height of conical cup (h) = 10 cm

\(\frac{r}{h} = \frac{20}{10} = 2\\r = 2h\)

Volume (V) of cone = \(\frac{1}{3}\pi r^2h\\\)

                                    \(\frac{1}{3}\pi (2h)^2h\\\)

                                     \(\frac{4}{3}\pi h^3\)

\(\frac{dV}{dt} = \frac{d}{dt}(\frac{4}{3}\pi h^3)\\\)

    =  \(4\pi h^2 \frac{dh}{dt}\)

Here \(\frac{dh}{dt} = 3cm, h = 4cm\)

\(\frac{dv}{dt}=4\pi \times4^2 \times 3\)

\(192\pi\) \(cm^3\)/sec

The rate at which water being poured into the cup when the water level is 4 cm is \(192\pi\) \(cm^3\)/sec

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Step-by-step explanation:

f(x)=2×2-32

f(5)=252-32

f(5)=220

Answer:

I pretty sure its C. OD-2) = 18

Step-by-step explanation:

\(~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] ~\dotfill\\\\ (4)^{\frac{-4}{2}} \implies (4)^{-2}\implies 4^{-2}\implies \cfrac{1}{4^2}\implies \cfrac{1}{16}\)

Answer:

  c:  8,20,25

Step-by-step explanation:

You want to know which set of side lengths can form a triangle from the sets ...

A set of lengths can form a triangle if and only if the sum of the shorter two exceeds the longest.

In sets a, b, d, the sum of the shorter two is equal to the longest. These lengths will form a line segment, not a triangle.

A triangle can be formed by ...

  c: 8,20,25 . . . . . . 8+20=28 > 25

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Answer:

3x=6

Step-by-step explanation:

From the question;

we are to state the three reasons why a function might be discontinuous at the point x=c

A funtion f(x) is discontinuous at a point x = c if

1.

2.

3.

If all this condition are statisfied, then the function is discontinuous at x = c

The function f(x).g(x) = 4x³ - 17x² + 15x + 18 and the domain of this function is (-∞, +∞).

The domain of a function is the set of all possible input values (also known as the independent variable) for which the function is defined.

In other words, it is the set of values of x for which the function has a corresponding output value (also known as the dependent variable).

Here we have

f(x) = 4x + 3

g(x) = x² - 5x + 6  

Then f(x) · g(x) can be calculated as follows

=> (4x + 3) [ x² - 5x + 6 ]

Using distributive property, we get:

=> 4x (x² - 5x + 6) + 3(x² - 5x + 6)

=> 4x³ - 20x² + 30x + 3x² - 15x + 18

=> 4x³ - 17x² + 15x + 18  

Hence, f(x).g(x) = 4x³ - 17x² + 15x + 18    

The domain of the function f(x) = 4x³ - 17x² + 15x + 18 is all real numbers since there are no restrictions or conditions that would make the function undefined for any value of x.

Therefore,

The function f(x).g(x) = 4x³ - 17x² + 15x + 18 and the domain of this function is (-∞, +∞).

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The values of the t statistic that are statistically significant at the α=0.005 level are t < -2.977 or t > 2.977.

To determine the values of the t statistic that are statistically significant at the α=0.005 level, we need to find the critical values of the t-distribution with 14 degrees of freedom (df = n - 1 = 15 - 1 = 14).

Using a t-table or a calculator, we can find that the critical values for a two-tailed test with α=0.005 are -2.977 and 2.977. This means that any t statistic less than -2.977 or greater than 2.977 would be considered statistically significant at the α=0.005 level.

In other words, if the calculated t statistic falls within the range of -2.977 to 2.977, we would fail to reject the null hypothesis (H0: μ=10) and conclude that there is not enough evidence to suggest that the population mean is different from 10. However, if the calculated t statistic falls outside of this range, we would reject the null hypothesis and conclude that there is evidence to suggest that the population mean is different from 10.

So, the values of the t statistic that are statistically significant at the α=0.005 level are t < -2.977 or t > 2.977.

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Answer:

The length of a side of the pool is 12 m²

Step-by-step explanation:

Let the area of the square plot be A. Since it is a square plot of 35 m side, its area is A = 35² = 1225 m².

Let A' be the area of the lawn. Since the area of the lawn is 1081 m², A' = 1081 m².

Now the area of the pool is the difference in area between the area of the square plot and the area of the lawn. So, let A'' be the area of the square swimming pool.

So, A'' = A - A'

A'' = 1225 m² - 1081 m²

A'' = 144 m².

Since the swimming pool is a square, the length of its side, L is thus

L = √A''

L = √144 m²

L = 12 m

So, the length of a side of the pool is 12 m.

Answer: Tre wants to mow 20 lawns in 5 days, so he needs to mow 20/5=<<20/5=4>>4 lawns per day.

If Tre can mow 3 lawns in 2 hours, then he can mow 1 lawn in 2/3=<<2/3=0.67>>0.67 hours.

This means Tre will have to work 4/0.67=<<4/0.67=6>>6 hours each day to mow all 20 lawns. Answer: \boxed{6}.

Step-by-step explanation:

Stephen's average speed for the round trip was 48 mph.

To find Stephen's average speed, we can use the formula:

Average Speed = Total Distance / Total Time

Let's assume the distance from Stephen's house to his aunt's house is 'd' miles.

On the way to his aunt's house, Stephen traveled at a speed of 60 mph. So the time taken for this leg of the trip is given by:

Time = Distance / Speed = d / 60

On the return trip, Stephen traveled at a speed of 40 mph. So the time taken for this leg of the trip is:

Time = Distance / Speed = d / 40

The total time for the round trip is the sum of the times for the outward and return trips:

Total Time = d / 60 + d / 40

To find the average speed, we divide the total distance by the total time:

Average Speed = Total Distance / Total Time

The total distance for the round trip is 2d (since it's the same roadway for both trips).

Average Speed = 2d / (d / 60 + d / 40)

Simplifying this expression, we get:

Average Speed = 2d / ((3d + 2d) / 120) = 2d / (5d / 120) = 2d * 120 / 5d = 240 / 5 = 48 mph

Therefore, Stephen's average speed for the round trip was 48 mph.

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Answer:

37/3

Step-by-step explanation:

Convert mixed numbers to improper fraction :

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▹ Answer

You can use a graphing calculator.

▹ Step-by-Step Explanation

Attached is a screenshot.

Hope this helps!

CloutAnswers ❁

━━━━━━━☆☆━━━━━━━

Answer:

See explanation and picture attached

Step-by-step explanation:

We can break down this expression into it's core components:

Since the constant here is -4, the y intercept is -4.

Since the value we are multiplying x by is \(\frac{2}{3}\), the slope is  \(\frac{2}{3}\). This means for every time we go horizontal 3 units, the line increases by 2.

The graph is attached.

Hope this helped!

Answer:

Step-by-step explanation:

Positive rational numbers will be in the right side of the number line.

Negative rational numbers will be in the right side of the number line.

18.256 will be lying in between 18.25 and 18.26

Total number of committees with 4 delegates from Nevada = 15 × C(24, 5)

Total number of committees with 5 delegates from Nevada = 6 × C(23, 4)

Total number of committees with 6 delegates from Nevada = 1 × C(22, 3)

What is Delegates?

An individual chosen to represent a group of people in a political assembly in the United States is known as a delegate.

To calculate the number of committees with at least 2 delegates from Nevada, we need to consider the number of possible combinations of delegates from five states: California, Arizona, New Mexico, Utah, and Nevada.

Since each state sends a team of 6 delegates, the total number of delegates available is 6 × 5 = 30 delegates.

Now let's count the number of committees with at least 2 delegates from Nevada. We can consider two scenarios:

Scenario 1: Exactly 2 delegates from Nevada

In this case, we need to select 2 delegates from Nevada and 7 delegates from the remaining 29 delegates (excluding the 2 from Nevada). The order of selection does not matter, so we use combinations.

Number of ways to choose 2 delegates from Nevada = C(6, 2) = 15 (using combinational formula)

Number of ways to select 7 delegates from the remaining 29 delegates = C(29, 7) = 3,138,225 (using combinational formula)

Total number of committees with exactly 2 delegates from Nevada = 15 × 3,138,225 = 47,073,375

Scenario 2: More than 2 delegates from Nevada

In this case, we can choose 3, 4, 5, or 6 delegates from Nevada. For each selection, we must select the remaining delegates from the remaining states.

Number of ways to select 3 delegates from Nevada = C(6, 3) = 20 (using combinational formula)

Number of ways to select 6 delegates from the remaining 27 delegates = C(27, 6) = 10,068,347 (using combinational formula)

Total Committees with 3 Nevada Delegates = 20 × 10,068,347 = 201,366,940

Similarly, we calculate the total number of committees with 4, 5, and 6 delegates from Nevada.

Number of ways to choose 4 delegates from Nevada = C(6, 4) = 15

Number of ways to choose 5 delegates from Nevada = C(6, 5) = 6

Number of ways to select 6 delegates from Nevada = C(6, 6) = 1

For each case, we calculate the number of ways to choose the remaining delegates from the remaining states using the combination formula.

Total number of committees with 4 delegates from Nevada = 15 × C(24, 5)

Total number of committees with 5 delegates from Nevada = 6 × C(23, 4)

Total number of committees with 6 delegates from Nevada = 1 × C(22, 3)

Calculating the above combinations will give you the final results.

Note: Calculations for remaining delegates from other states are required to account for the different number of delegates in each scenario.

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The definition of an eigenvector of A with eigenvalue λ.

True.

By definition, an eigenvector of a matrix A is a non-zero vector x that satisfies the equation Ax = λx, where λ is a scalar called the eigenvalue corresponding to x.

So if Ax = λx for some scalar λ, then x is a non-zero vector that satisfies the definition of an eigenvector of A with eigenvalue λ.

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Hey there!

f(x) = g(x) = h(x) = y

Since, you have the value of x (8), you can substitute it in the equation to make it easier to solve. We just have to find your f(x) value (also known as the y-value)

f(x) -0 -3 * x^2 - 2

f(x) = -3x^2 - 2

y = -3x^2 - 2

y = -3(8)^2 - 2

y = -3(8^2) - 2

y = -3(8 * 8) - 2

y = -3(64) - 2

y = -192 - 2

y = -194

Therefore, your answer should be:

f(x) = -194

Good luck on your assignment & enjoy your day!

~Amphitrite1040:)

Answer:

3.25 pounds

Step-by-step explanation:

x + 4 (1/2) + 3 (1/2) = 11

x + 4.25 + 3.5 = 11

x + 7.75 = 11

x = 11 - 7.75

x = 3.25 pounds