Homework 14: Max/Min Problems

Part A: Basic Problems

code for piecewise f(x)

Exercise 14. 1 : Let [Maple Math]

(a) Find the maximum value of f(x) for x between 0 and 4.

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(b) Find the minimum value of f(x) for x between 0 and 4.
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(c) Graph f(x) to check your answers.

[Maple Plot]

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Exercise 14.2. Find the dimensions of the rectangle of largest area having fixed perimeter P. (Compare with Example 4 above.)
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Exercise 14.3 : A box with square base and no top is to hold a volume [Maple Math] . Find (in terms of [Maple Math] ) the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base. (This ratio will not involve [Maple Math] .)

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Exercise 14.4. You have L feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest size possible (in [Maple Math] ) for the play area?
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Exercise 14.5 . Marketing tells you that if you set the price of an item at $10 then you will be unable to sell it, but that you can sell 500 items for each dollar below $10 you set the price. Suppose your fixed costs total $3000, and your marginal cost is $2 per item. What is the most profit you can make?
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Exercise 14.6: Find the area of the largest rectangle that fits inside a semicircle of radius r (one side of the rectangle is along the diameter of the semicircle).
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Exercise 14.7 . For a cylinder with given surface area S, find the ratio of height to base radius that maximizes the volume.
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Exercise 14.8: You want to make cylindrical containers of a given volume [Maple Math] using the least amount of construction material. The lateral side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side [Maple Math] , so that [Maple Math] of material is needed (rather than [Maple Math] , which is the total area of the top and bottom). Find the optimal ratio of height to radius.

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Exercise 14.9. Given a right circular cone, you put an upside-down cone inside it so that its vertex is at the center of the base of the larger cone and its base is parallel to the base of the larger cone. If you choose the upside-down cone to have the largest possible volume, what fraction of the volume of the larger cone does it occupy? (Let [Maple Math] and [Maple Math] be the height and base radius of the larger cone, and let [Maple Math] and [Maple Math] be the height and base radius of the smaller cone. Hint: Use similar triangles to get an equation relating [Maple Math] and [Maple Math] .)

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Exercise 14.10 .
In Example 9, what happens if [Maple Math] (i.e., your speed on sand is greater than your speed on the road)?

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Exercise 14.11 . A container holding a fixed volume is being made in the shape of a cylinder with a hemispherical bottom. (The hemispherical bottom has the same radius as the cylinder.) Find the ratio of height to radius of the cylinder which minimizes the cost of the container if (a) the cost per unit area of the bottom is twice as great as the cost per unit area of the side, and the container is made with no top; (b) the same as in (a), except that the container is made with a circular top, for which the cost per unit area is 1 1/2 times the cost per unit area of the side.
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Exercise 14.12. (a) A square piece of cardboard of side a is used to make an open-top box by cutting out a small square from each corner and bending up the sides. How large a square should be cut from each corner in order that the box have maximum volume? (b) What if the piece of cardboard used to make the box is a rectangle of sides a and b ? Note that the side [Maple Math] of the square to be cut from each corner must be less than a /2 and b /2, and in this way you can eliminate one of the two solutions of [Maple Math] . In addition, use the second derivative test to be sure your value of [Maple Math] really gives the maximum volume.

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Exercise 14.13 . A window consists of a rectangular piece of clear glass with a semicircular piece of colored glass on top. Suppose that the colored glass transmits only [Maple Math] times as much light per unit area as the clear glass ( [Maple Math] is some fraction). If the distance from top to bottom (across both the rectangle and the semicircle) is fixed, find (in terms of [Maple Math] ) the ratio of vertical side to horizontal side of the rectangle for which the window lets through the most light. For what range of values of [Maple Math] does this question make sense?
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Exercise 14.14 . In the same situation as Problem 4, suppose that instead of a rectangular shape you decide to make the play area in the form of a trapezoid, using three sections of fence of length l/3 each (see the diagram). Find the angle [Maple Math] that the sides should be turned outward in order for the enclosed area to be maximal, and find the maximum play area of this form. Is the maximum play area greater or less than in Problem 4?

code for diagram

[Maple Plot]

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Part B: Additional Practice: (do these as soon as possible, certainly before the final exam):

Exercise 14.15. A terrorist sees a plane flying away from him at 500 mph at a height of 2 miles. At time t = 0 it is flying over a point on the ground which is 1.5 miles from him. At that instant he shoots a missile at the plane, but because of nervousness and perhaps a subconscious sense of morality, he mistakenly aims directly at the present location of the plane, rather than slightly ahead. For this reason he misses the plane. The purpose of this problem is to determine by how much. Suppose the missile travels at 2000 mph, and ( neglect both air resistance and gravity ). First find parametric equations for both the plane and the missile. Then find the time at which the two objects are closest. ( Hint : Use the distance formula, and save yourself unnecessary algebra by finding the time at which the square of the distance is minimal.) By how much does the missile miss the plane?
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Exercise 14.16. You are designing a poster to contain a fixed amount [Maple Math] of printing (measured in square inches) and have margins of a inches at the top and bottom and b inches at the sides. Find the ratio of vertical dimension to horizontal dimension of the printing on the poster if you want to minimize the amount of posterboard needed.

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Exercise 14.17 . The strength of a rectangular beam is proportional to the product of its width times the square of its depth. Find the dimensions of the strongest beam that can be cut from a cylindrical log of radius [Maple Math] . \vfill





Exercise 14.18 . What fraction of the volume of a sphere is taken up by the largest cylinder that can be fit inside the sphere?
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Exercise 14.19 . The U.S. post office will accept a box for shipment only if the sum of the length and girth (distance around) is at most 108 in. Find the dimensions of the largest acceptable box with square front and back.
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Exercise 14.20 : Find the dimensions of the lightest cylindrical can containing 0.25 liter ( = 250 [Maple Math] ) if the top and bottom are made of a material that is twice as heavy (per unit area) as the material used for the side.
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Exercise 14.21. A conical paper cup is to hold a fixed volume of water. Find the ratio of height to base radius of the cone which minimizes the amount of paper needed to make the cup. Use the formula [Maple Math] for the area of the side of a cone.
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Exercise 14.22 . If you fit the cone with the largest possible surface area (lateral area plus area of base) into a sphere, what percent of the volume of the sphere is occupied by the cone? Compare your answer with the answer to Example 7.
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Exercise 14.23. Two electrical charges, one a positive charge [Maple Math] of magnitude a and the other a negative charge [Maple Math] of magnitude [Maple Math] , are located a distance [Maple Math] apart. A positively charged particle [Maple Math] is situated on the line between [Maple Math] and [Maple Math] . Find where [Maple Math] should be put so that the pull away from [Maple Math] towards B is minimal. Here assume that the force from each charge is proportional to the strength of the source and inversely proportional to the square of the distance from the source.

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Exercise 14.24 . Find the fraction of the area of a triangle that is occupied by the largest rectangle that can be drawn in the triangle (with one of its sides along a side of the triangle). Show that this fraction does not depend on the dimensions of the given triangle.
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Exercise 14.25 . How are your answers to Problem 5 affected if the cost per item for the [Maple Math] items, instead of being simply 2, decreases below 2 in proportion to [Maple Math] (because of economy of scale and volume discounts) by 1 cent for each 25 items produced.
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Exercise 14.26 . You are standing near the side of a large wading pool of uniform depth when you see a child in trouble. You can run at a speed [Maple Math] on land and at a slower speed [Maple Math] in the water. Your perpendicular distance from the side of the pool is a , the child's perpendicular distance is b , and the distance along the side of the pool between the closest point to you and
the closest point to the child is c (see the drawing).

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[Maple Plot]


Without even stopping to do any calculus, you instinctively choose the quickest route (shown in the diagram at the right) and save the child. Our purpose is to derive a relation between the angle [Maple Math] your path makes with the perpendicular to the side of the pool when you're on land, and the angle [Maple Math] your path makes with the perpendicular when you're in the water. To do this, let [Maple Math] be the distance between the closest point to you at the side of the pool and the point where you enter the water. Write the total time you run (on land and in the water) in terms of [Maple Math] (and also the constants [Maple Math] ). Then set the derivative equal to zero. The result, called ``Snell's law'' or the ``law of refraction,'' also governs the bending of light when it goes into water.

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