Calculus AB and Calculus BC

CHAPTER 4 Applications of Differential Calculus

M. RELATED RATES

If several variables that are functions of time t are related by an equation, we can obtain a relation involving their (time) rates of change by differentiating with respect to t.

EXAMPLE 36

If one leg AB of a right triangle increases at the rate of 2 inches per second, while the other leg AC decreases at 3 inches per second, find how fast the hypotenuse is changing when AB = 6 feet and AC = 8 feet.

Image

FIGURE N4–21

SOLUTION: See Figure N4–21. Let u, v, and z denote the lengths respectively of AB, AC, and BC. We know that Image Since (at any time) z2 = u2 + v2, then

Image

At the instant in question, u = 6, v = 8, and z = 10, so

Image

EXAMPLE 37

The diameter and height of a paper cup in the shape of a cone are both 4 inches, and water is leaking out at the rate of Image cubic inch per second. Find the rate at which the water level is dropping when the diameter of the surface is 2 inches.

SOLUTION: See Figure N4–22. We know that Image and that h = 2r.

Here, Image

Image at any time.

When the diameter is 2 in., so is the height, and Image The water level is thus dropping at the rate of Image in./sec.

Image

FIGURE N4–22

EXAMPLE 38

Suppose liquid is flowing into a vessel at a constant rate. The vessel has the shape of a hemisphere capped by a cylinder, as shown in Figure N4–23. Graph y = h(t), the height (= depth) of the liquid at time t, labeling and explaining any salient characteristics of the graph.

Image

FIGURE N4–23

SOLUTION: Liquid flowing in at a constant rate means the change in volume is constant per unit of time. Obviously, the depth of the liquid increases as t does, so h (t) is positive throughout. To maintain the constant increase in volume per unit of time, when the radius grows, h (t) must decrease. Thus, the rate of increase of h decreases as h increases from 0 to a (where the cross-sectional area of the vessel is largest). Therefore, since h (t) decreases, h (t) < 0 from 0 to a and the curve is concave down.

As h increases from a to b, the radius of the vessel (here cylindrical) remains constant, as do the cross-sectional areas. Therefore h (t) is also constant, implying that h(t) is linear from a to b.

Note that the inflection point at depth a does not exist, since h (t) < 0 for all values less than a but is equal to 0 for all depths greater than or equal to a.

BC ONLY