box deep. In either case the volume will be small. Somewhere between these two extremes will be a box whose volume is greater than that of any other, that is, a box of maximum volume. Let x be the side of the square cut out. x, and the side of the base is 4 - 2x. function of x is

Then the depth of the box is Hence the volume, expressed as a

The graph of V is readily plotted. From it, the value of x which makes V a maximum appears to be somewhat less than unity. By computing V for a large number of values of x, we could approximate the best value of x. By means of the derivative we can avoid this labor, and obtain the exact value.

υλ

12

3 x

FIG. 166.

At the maximum point the tangent line is horizontal, and hence its slope is zero, so that D&V = 0.

We therefore compute the derivative, set it equal to zero, and find the value of x which produces this result. We then have

or, dividing by 4,

Factoring

and hence

=

=

3.x2 8x + 4

=

(3x-2) (x − 2) = 0,

=

x = 2/3 or 2.

2

From the figure, we see that x makes V a maximum, and x = gives a minimum. That x 2 gives a minimum follows also from the fact that if x 2, then V = 0, that is, the box will not hold anything. The preliminary discussion showed the existence of a maximum, and as x = is the only other possibility it must be the value of x for which V is a maximum. Either of the criteria in Section 96 may also be applied to show that V is a maximum if x = 3.

In order then, to have a box of maximum capacity, we must cut out squares from the corners of an inch on a side. The depth of the box will be of an inch, the side of the base will be 4 - 2 × 3 = 23 inches, and the capacity will be V (23)2=422 cu. in.

=

EXAMPLE 2. As large a rectangular stick of timber as possible is to be sawed from a log 10 inches in diameter at the smaller end, the length of the stick to be the same as that of the log. Find its other dimensions.

Let V denote the volume of the stick of timber, I its length, and A the area of an end. Then V = lA, and since l is constant, the volume V will be a maximum if and only if A is a maximum. The end of the stick is a rectangle inscribed in a circle whose diameter is 10, and the problem reduces to a determination of the dimensions of the maximum rectangle which can be inscribed in this circle.

The area of the rectangle is

A

=

xy,

(1)

which involves the two variable dimensions x and y. These are connected

In order to find the derivative of A we write it in the form

(2)

(3)

lie between these extremes. Hence as x increases from 0 to 10, A first increases and then decreases, and as A does not become infinite it therefore has a maximum in this interval. But x = 5 V2 is the only point in this interval at which the tangent to the graph of A is horizontal, and it must therefore give the maximum value.

=

The corresponding value of y, from (2), is y 5 √2. Hence the end of the largest stick of timber will be a square whose side is 5 V2 inches, or very nearly 7 inches.

EXERCISES

1. A box is to be made by cutting squares from the corners of a piece of cardboard 6 by 8 inches, and folding up the sides. Find the dimensions if the capacity is to be a maximum.

2. A chicken yard is to be made from 36 feet of poultry fencing, the side of a barn being used for one side of the yard. Find the dimensions in order that the yard may be as large as possible.

3. Find the dimensions and capacity of the largest box which can be made with a square base and no top if the total amount of cardboard in the box is 48 square inches.

4. What should be the dimensions of a rectangular garden plot with a perimeter of 12 rods, in order to have the greatest area possible?

5. A two acre pasture in the form of a rectangle is to be fenced off along the bank of a straight river, no fence being needed along the river. Find the dimensions, in rods, in order that the fence may cost as little as possible. 6. The legs of an isosceles triangle are 6 inches long. How long must the base be in order that the area may be a maximum?

7. The height of a rifle ball fired vertically upward with an initial velocity of 1200 feet per second is s = 1200t16t2. How high will it rise?

8. By the Parcel Post regulations, the combined length and girth of a package must not exceed 6 feet. Find the dimensions and volume of the largest parcel which can be sent in the shape of a box with square ends.

9. A farmer has 150 rods of fencing. Find the dimensions and area of the largest rectangular field he can enclose and divide into two equal parts by a fence parallel to two of the sides.

10. If the total area of the field in the preceding Exercise is to be 150 square rods, find the dimensions and the amount of fencing needed if the latter is to be a minimum.

11. A rectangular cistern is to be built with a square base and open top. Find the proportions if the amount of material used is to be a minimum.

12. A rectangular piece of ground is to be fenced off and divided into three equal parts by fences parallel to one of the sides. What should the dimensions be in order that as much ground as possible may be enclosed with 16 rods of fence?

13. If the total area enclosed in Exercise 12 is an acre, find the dimensions in order that the total length of the fence should be a minimum.

14. The number of tons of coal consumed per hour by a certain ship is 0.3 + 0.001 V3, where V is the speed in knots. For a voyage of 1000 knots at V knots per hour, find the total consumption of coal. For what speed is the consumption of coal least?

15. Divide a string 16 inches long into two parts, so that the combined area of the square and circle with perimeters equal to the parts shall be a minimum.

16. Find the coördinates of the maximum or minimum point of y x2 + bx + c.

=

17. At what height should a light be placed above a writing table in order that a small portion of the table, at a given horizontal distance d from the point directly below the light, may receive the greatest illumination possible? (It is known that the intensity of illumination varies inversely as the square of the distance and directly as the sine of the angle between the line from the light to the point in question and the table. Express the illumination as a function of the height of the light, and find the value of the height which gives the maximum illumination.)

18. A rectangular box is to be made by cutting out the corners of a rectangular piece of cardboard and folding up the sides. If the depth of the box is to be 2 inches, find the dimensions of the smallest piece of cardboard which will make a box to contain 72 cubic inches.

102. Related Rates. If two variables x and y are functions of the time t, then their derivatives with respect to t, Dix and D1y, measure the rates at which x and y are changing. These rates will be related, that is, connected by a relation, if y is a function of x.

For example, if x and y are functions of t such that y = x3, then the rates at which x and y change satisfy the equation Diy 3x2Dix, which is obtained by differentiating the given equation with respect to t. The rate of change of y depends on the value of x as well as on that of the rate of change of x.

=

In solving problems involving the rates at which two variables change, it is first necessary to determine the relation y = f(x) connecting the variables. Then the relation between the rates of change of x and y is found by differentiating y = f(x) with respect to t.

EXAMPLE 1. Oil dropped on a smooth floor spreads out in the form of a circle. If the radius is increasing at the rate of an inch per second when it is 6 inches long, how fast is the area increasing?

If r denotes the radius and A the area, the question asked may be expressed symbolically as follows: If Dr when r = 6, what is the value of DA?

=

In order to answer this question we must have a relation between D‚A and Dr. And to obtain this relation we must first express A as a function From plane geometry we have

of r.

Differentiating both sides with respect to t, we get

DA = 2πrDr.

Substituting the values of π, r, and Dr, the oil covered area is increasing at the rate of D1A = 2 × 22/7 × 6 × 1 = 18.85 square inches per second. EXAMPLE 2. A man walks along a sidewalk at the rate of three miles an hour (4.4 feet per second), approaching a house which stands back 7 feet from the walk. When he is 24 feet from the

7

У

FIG. 168.

walk leading to the house, how rapidly is he approaching the house?

The rate at which he approaches the house is the derivative of his distance from the house with respect to time. If y denotes his distance from the house, we seek Dy.

The rate at which the man walks is the rate of change of his distance from some point on the walk with respect to the time. This point is conveniently chosen as the point on the sidewalk directly in front of the house, because we know this distance at the time we wish to determine Dy. Let his distance from this point be x. Then Dix = 4.4.

To find the relation between these two derivatives we must first express y as a function of x. From the figure we have, at any time,

and hence, by Theorem 5, page 270, differentiating with respect to t,

We are told that when x = 24, Dix = 4.4, and hence,

1. In Example 1 above, find the rate at which the circumference is increasing when the radius is 6 inches. Find also the rate at which the area and circumference are increasing when the radius is 10 inches. What essential difference is there in the rates at which the circumference and area increase?

2. Imagine a belt stretched around the earth at the equator (radius 3963 miles). If the radius of the belt increases uniformly at the rate of 3 feet per second, how much will the circumference increase in the first second?