Higher Concepts of Mathematics
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.
Using the formula for the area of a circle,
take the derivative of both sides of this equation with respect to time t.
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.
Thus, at a radius of 4 feet, the area is increasing at a rate of 12.6 square feet per
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?
Start with the Pythagorean Theorem for a right triangle:
a2 = c2 - b2