Higher Concepts of Mathematics
CALCULUS
Example 1:
A stone is dropped into a quiet lake, and waves move in circles outward from the
location of the splash at a constant velocity of 0.5 feet per second. Determine the
rate at which the area of the circle is increasing when the radius is 4 feet.
Solution:
Using the formula for the area of a circle,
A pr2
take the derivative of both sides of this equation with respect to time t.
dA
dt
2pr
dr
dt
But, dr/dt is the velocity of the circle moving outward which equals 0.5 ft/s and
dA /dt is the rate at which the area is increasing, which is the quantity to be
determined. Set r equal to 4 feet, substitute the known values into the equation,
and solve for dA /dt.
dA
dt
2pr
dr
dt
dA
dt
(2)(3.1416)(4 ft)
0.5 ft/s
dA
dt
12.6 ft2/s
Thus, at a radius of 4 feet, the area is increasing at a rate of 12.6 square feet per
second.
Example 2:
A ladder 26 feet long is leaning against a wall. The ladder starts to move such
that the bottom end moves away from the wall at a constant velocity of 2 feet per
second. What is the downward velocity of the top end of the ladder when the
bottom end is 10 feet from the wall?
Solution:
Start with the Pythagorean Theorem for a right triangle:
a2 = c2 - b2
Rev. 0
Page 39
MA-05
