A rancher has 300 feet of fencing available to enclose ...



MATH 2053 PRACTICE TEST 3

1. A rancher has 300 feet of fencing available to enclose three adjacent rectangular corrals as shown in the diagram. (The fence goes around the outside and between the corrals.) What dimensions should be used so the total enclosed area is a maximum?

Let the dimensions be x by y. Then you have a picture

The problem is to maximize [pic]

You need to find a relationship between the variables. The sum of the lengths of all of the sides is 300 feet so,

[pic]

Divide each term by 2

[pic]

You have to solve for either x or y. In this case it is easier to solve for x.

[pic]

[pic]

[pic]

[pic] if [pic]

[pic]

Then [pic]

Note that [pic]so we have found a maximum.

Answer: The dimensions are 75 feet by 37.5 feet.

2. A rectangular page is to contain 32 square inches of print. The margins on each side are one inch and the margins on the top and bottom are two inches. Find the dimensions of the page such that the least amount of paper is used.

Draw a picture. The relationship between x and y comes from the printed area being 32 square inches. You should choose variables so that the relationship between them is as simple as possible. Make the printed area x inches by y inches. Then the dimensions of the page are [pic] by [pic]

Minimize [pic] with [pic], or [pic]

[pic]

[pic]

[pic] if [pic]

[pic]

Since x cannot be negative, [pic]. Then [pic]

The question asks for the dimensions of the page, which are x + 4 by y + 2.

Note that [pic]so we have found a minimum

The dimensions are 12 inches by 6 inches

3. Starting with [pic] apply two iterations of Newton’s method to approximate a zero of [pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Note that if you have [pic] on the display of your calculator you can use the ans key to enter [pic].

[pic]

4. Evaluate [pic]and [pic] if [pic]; [pic]; [pic]

[pic]

[pic]

5. Use differentials to approximate [pic]

Use [pic]

The idea is to choose an appropriate function, in this case [pic] and reasonable values of x and [pic]such that the right hand side is easy to evaluate and[pic] is small

[pic]

[pic]

Then [pic]

Choose [pic] and [pic], then

[pic]

6. Evaluate the indefinite integrals.

(a) [pic] (b) [pic] (c) [pic]

[pic] [pic] =[pic]

[pic]

7. Solve the differential equation [pic]

[pic]

You now use the fact that [pic] to find c.

[pic]

Answer: [pic]

8. Use the limit process to find the area of the region between the graph of the function and the x-axis over the given interval. [pic]

Divide [0, 2] into n equal subintervals of width [pic]

[pic].

You need to evaluate [pic]

Take [pic] to be the right hand edge of edge of the ith interval then [pic]

[pic]

[pic]

9. Write the limit as a definite integral on the interval [4, 7]: [pic]

A Riemann sum is of the form [pic] and [pic]= [pic]

Answer: [pic]

10. Given that [pic] find

(a) [pic] (b) [pic] (c)[pic]

(4 0 [pic]

[pic]

11. For certain commodity the demand price is [pic]. The cost function is [pic]. Find:

(a) The level of production that maximizes profit.

Revenue = [pic]

Profit = Revenue ( cost

[pic]

Profit is maximum when [pic]

[pic]

Note that this gives a maximum because [pic]

(b) The maximum profit

[pic]

Note that a negative maximum profit means that you will make a loss no matter what.

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x

x

y

y

y

y

x

y

x + 4

y + 2

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