Page 74 - Calc for the Clueless

A girl flying a kite plays out a string at 2 feet per second. The kite moves horizontally at an altitude of 100 feet.
If there is no sag in the string, find the rate at which the kite is moving when 260 feet of string have been played
out.

We first have to imagine a right triangle with the vertical distance, the horizontal distance, and the string. We
then must determine which are the constants and which are the variables. The only constant is that the kite is
always 100 feet high. (I wonder how they keep the kite exactly 100 feet high?)
Although we are given the length of the string, it is only at a particular instant. The string length, s, and
horizontal length, x, are changing.

2
2
2
The equation is x + 100 = s . Differentiating with respect to time, we get

ds/dt is 2. Since s = 260, the Pythagorean theorem tells us that x = 240 feet. (You should have memorized the
common Pythagorean triples. This is a 5,12,13 right tri angle, or more precisely, a 10,24,26 right triangle.)
Substituting in

we get

Example 20—

Sand is leaking from a bag and is forming a cone in which the radius is 6 times as large as the height. Find the
rate at which the volume is increasing when the radius is 3 inches and the height is increasing at 2 inches per
minute.

We know that the volume of the cone is V = (1/3)πr h. There are two variables, but we also know that r = 6h.
2
We can write V in terms of one unknown, but which unknown? Since dh/dt = 2 is given, let us write everything
in terms of h.

Example 21—

When a gas is compressed adiabatically (with no gain or loss of heat), it satisfies the formula PV = k; k is a
1.4
constant. Find the rate at which the pressure is changing if the pressure P is 560 pounds per square inch, the
volume is 70 cubic inches, and the volume is increasing at 2 cubic inches per minute.