5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 14:46

Applications of Derivatives 189

4. Using your calculator, find the shortest distance between the point (4, 0) and the line
   y=x. (See Figure 9.3-1.)

[–6.3, 10] by [–2, 6]

Figure 9.3-1

Answer:

S=

√

(x−4)^2 +(y−0)^2 =

√

(x−4)^2 +x^2

Entery 1 =

√

(x−4)^2 +x^2 andy 2 =d(y 1 (x),x).

Use the [Zero] function fory 2 and obtainx=2. Note that whenx<2,y 2 <0, which

meansy 1 is decreasing and whenx>2,y 2 >0, which meansy 1 is increasing, and

thus atx=2,y 1 is a minimum. Use the [Value] function fory 1 atx=2 and obtain

y 1 = 2 .82843. Thus, the shortest distance is approximately 2.828.

9.4 Practice Problems

Part A The use of a calculator is not

allowed.

1. A spherical balloon is being inflated. Find
   the volume of the balloon at the instant
   when the rate of increase of the surface area
   is eight times the rate of increase of the
   radius of the sphere.


2. A 13-foot ladder is leaning against a wall. If
   the top of the ladder is sliding down the
   wall at 2 ft/sec, how fast is the bottom of
   the ladder moving away from the wall when
   the top of the ladder is 5 feet from the
   ground? (See Figure 9.4-1.)

13 ft

Wall

Ground

Figure 9.4-1

3. Air is being pumped into a spherical balloon
   at the rate of 100 cm^3 /sec. How fast is the
   diameter increasing when the radius is 5 cm?


4. A woman 5 feet tall is walking away from a
   streetlight hung 20 feet from the ground at
   the rate of 6 ft/sec. How fast is her shadow
   lengthening?


5. A water tank in the shape of an inverted
   cone has a height of 18 feet and a base
   radius of 12 feet. If the tank is full and the
   water is drained at the rate of 4 ft^3 /min,
   how fast is the water level dropping when
   the water level is 6 feet high?


6. Two cars leave an intersection at the same
   time. The first car is going due east at the
   rate of 40 mph and the second is going due
   south at the rate of 30 mph. How fast is the
   distance between the two cars increasing
   when the first car is 120 miles from the
   intersection?


7. If the perimeter of an isosceles triangle is
   18 cm, find the maximum area of the
   triangle.