Word Problems Examples

A snail travels with velocity v(t) = 2 + 0.5sin t feet per second, where t is given in seconds. If the snail starts traveling at noon (t = 0), what does the expression

represent in the context of this problem?

When given an integral and asked to figure out what it means or what it represents, it's a good idea to first determine the units of the integral and the units of the variable of integration.

We're told that v(t) is measured in feet per second and t is measured in seconds. This means

is measured in feet. We're told that t = 0 corresponds to noon. Since t is measured in seconds, t = 60 seconds corresponds to 12:01pm.

Let's put all this information into a complete sentence. The integral

is the distance in feet the snail travels from 12:00 noon to 12:01 pm.

When you're told to explain what an integral is "in the context of the problem" or "in the language of the problem" or "in terms of the problem," we recommend first figuring out the units of the integral and the units of the variable of integration. The integral tells you how much a certain quantity has changed, and the limits of integration tell you the starting and ending points between which this change happens.

Water pressure p at a depth of h feet below the surface of the water is given by the formula

p(h) = 62.4h

measured in "pcf," or "pounds per cubic foot."

Evaluate the integral

and explain what it means in terms of this problem.

We know h is measured in feet. Since p'(h) is the rate of change of p(h), the integral of p'(h) is measured in the same units as p(h). In this case the units are pcf (pounds per cubic foot).

Using the FTC,

This is the change in pressure from h = -15 to h = 0. Including units,

is the change in pressure as one moves from a depth of 15 feet below the surface of the water up to the surface of the water.

Sonya can paint at a rate of

v(t) = 150 – 4t

square feet per hour, where t is the number of hours since she started painting. How long will it take Sonya to paint 568 square feet?

Over H hours Sonya can paint

square feet. We want to know how many hours it will take her to paint 568 square feet—that is, we want to know the value of H for which

We need to solve this integral equation for the variable H. First, we evaluate the integral:

Now we can work with an equation that doesn't involve integrals. Dividing through by 2, we need to solve

H² – 75H + 284 = 0.

By factoring or using the quadratic formula, we get

H = 71, H = 4.

It's unreasonable to expect that Sonya could paint for 71 hours straight, so we take H = 4 as our answer. It will take Sonya 4 hours to paint 568 square feet.

A pan of brownies has been prepared at room temperature, 70°F, and is put into an oven that has been pre-heated to 350°F. When the brownies have been in the oven for t minutes they heat up at a rate inversely proportional to (t + 1), with proportionality constant k. If the brownies are 250 degrees after being in the oven for 15 minutes, what is k?

Let T(t) be the temperature of the brownies after t minutes in the oven. The problem tells us that T(0) = 70 and that T(15) = 250. Since the brownies heat at a rate inversely proportional to (t + 1) with constant of proportionality k, we know

We also know that the change in temperature from t = 0 to t = 15 is

There are two ways to evaluate this integral. We can put in the formula for T'(t), find an antiderivative, and evaluate:

Or we can use the FTC and the fact that we're integrating a rate of change:

Since the integral must come out the same whichever way we evaluate it, we must have

k ln 16 = 180

so the constant of proportionality is