## 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.

**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 Since (at any time) *z*^{2} = *u*^{2} + *v*^{2}, then

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

**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 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 and that *h* = 2*r*.

Here,

at any time.

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

**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.

**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**