Word Problems - Rates of Change

This lesson centers around solving Word Problems pertaining to Average Rate of Change and Instanteneous Rate of Change. Here are the formulas for the respective Rates of Change:


For additional information and practice with these formulas, review Slope of a Secant and Slope of a Tangent!!


It's important to identify what Rate of Change you're calculating to determine which formula to use. If the question is asking for Average Rate of Change, use the secant formula. If it's asking for the Instantaneous Rate of Change, use the tangent formula.

When calculating Rates of Change, it's good to use variables to represent different quantities. Also, remember to include units in your final answer!


The Surface Area of a snowball, in square centimeters, is modelled by the equation \(S = 4\pi r^2\) where \(r\) is the radius, in centimeters. The Volume, in cubic centimeters, is modelled by \(V = \cfrac{4}{3}\pi r^3\).

  1. Determine the average rate of change of the surface area as the radius changes from \(20\;[\text{cm}]\) to \(25\;[\text{cm}]\).
  2. Determine the instantaneous rate of change of volume when the radius is \(10\;[\text{cm}]\).

An outdoor hot tub holds \(2700\;[\text{L}]\) of water. The volume of water in the tub is modelled by the function \(V(t) = \cfrac{1}{12}(180-t)^2\), where \(V\) is the volume of water in the hot tub, in litres, and \(t\) is the time, in minutes, that the valve is open. Determine the instantaneous rate of change in the volume of water in \(60\;[min]\).

Show that the rate of change in the Volume of a cube with respect to its edge length is equal to half the Surface Area of the cube.

Determine the equation of the line that is perpendicular to the tangent of \(y = x^5\) at \(x=-2\) and which passes through the tangent point.

Review these lessons:
Try these questions: