Applications of Derivatives 161- Using your calculator, find the shortest distance between the point (4, 0) and the line
y=x. (See Figure 8.3-1.)
[–6.3,10] by [–2,6]
Figure 8.3-1
Answer:
S=√
(x−4)^2 +(y−0)^2 =√
(x−4)^2 +x^2
Entery 1 =((x−4)∧ 2 +x∧2)∧(.5) andy 2 =d(y1(x),x).
Use the [Zero] function fory2 and obtainx=2. Note that whenx<2,y 2 <0 and
whenx>2,y 2 >0, which means atx=2,y1 is a minimum. Use the [Value]
function fory1atx=2 and obtainy 1 = 2 .82843. Thus, the shortest distance is
approximately 2.828.8.4 Practice Problems
Part A The use of a calculator is not
allowed.- 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. - 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 8.4-1.)
13 ftWallGround
Figure 8.4-13.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.