Math 1325
Section 14.5 – Related Rates
Related Rates
It is common for variables to be functions of time. Time is often present implicitly in a
mathematical model, meaning that the derivatives with respect to time must be found
using the method of implicit differentiation.
Suppose that x and y are both functions of t , which can be considered to represent t ,
and that x and y are related by the equation
xy 2 + y = x 2 + 17 .
Suppose further that when x = 2 and y = 3 , then dx dt = 13 . Find the value of dy dt at
that moment.
Assume x and y are functions of t . Evaluate
3 xe y =
9 − ln 729 + 6 ln x
dy
for the following.
dt
dx
= 6 , x = 3, y = 0
dt
1
Assume x and y are functions of t . Evaluate
6t
y ln x + xe y =
dy
for the following.
dt
dx
= 5 , x =1, y = 0
dt
Solving a Related Rate Problem
1. Identify all given quantities, as well as the quantities to be found. Draw a sketch
when possible.
2. Write an equation relating the variables of the problem.
3. Use implicit differentiation to find the derivative of both sides of the equation in
Step 2 with respect to time. (Remember, you cannot plug-in any values until after
the derivative is taken.)
4. Solve for the derivative giving the unknown rate of change and substitute the
given values.
A small rock is dropped into a lake. Circular ripples spread over the surface of the water,
with the radius of each circle increasing at the rate of 3 2 ft per second. Find the rate of
change of the area inside the circle formed by a ripple at the instant the radius is 4 ft.
Recall, the formula for the area of a circle is A = π r 2 . Leave your answer in terms of π .
2
A 50-ft ladder is placed against a large building. The base of the ladder is resting on an
oil spill, and it slips away from the building at a rate of 3 ft per minute. Find the rate of
change of the height of the top of the ladder above the ground at the instant when the base
of the ladder is 30 ft from the base of the building.
Use the Pythagorean Theorem: a 2 + b 2 =
c2 .
A cone-shaped icicle is dripping from the roof. The radius of the icicle is decreasing at a
rate of 0.2 cm per hour, while the length is increasing at a rate of 0.8 cm per hour. If the
icicle is currently 4 cm in radius and 20 cm long, is the volume of the icicle increasing or
decreasing, and at what rate?
1
Recall, the formula for the volume of a cone is V = π r 2 h . Leave your answer in terms
3
of π .
3
A sand storage tank used by the highway department for winter storms is leaking. As the
sand leaks out, it forms a conical pile. The radius of the base of the pile increases at the
rate of 1 inch per minute. The height of the pile is always twice the radius of the base.
Find the rate at which the volume of the pile is increasing at the instant the radius of the
base is 5 inches.
1
Recall, the formula for the volume of a cone is V = π r 2 h . Leave your answer in terms
3
of π .
4
A pulley is on the edge of a dock, 8 ft above the water level. A rope is being used to pull
in a boat. The rope is attached to the boat at water level. The rope is being pulled in at
the rate of 1 ft per second. Find the rate at which the boat is approaching the dock at the
instant the boat is 8 ft from the dock.
Use the Pythagorean Theorem: a 2 + b 2 =
c2 .
5
Find the rate of change of total revenue, cost, and profit with respect to time. Assume
R ( x ) and C ( x ) are in dollars given that R (=
x ) 40 x − 0.5 x 2 , C ( x=
) 6 x + 15 , when
x = 30 and dx dt = 15 units per day.
6