Bob leaves for a trip at 3pm (time t = 0) and drives with velocity miles per hour, where t is measured in hours. Find
and explain what it represents in the language of the problem.
First we find the value of the integral:
To explain what the integral represents in the language of the problem, we need to know units for t and units for the integral. We're told that t is measured in hours and v(t) in miles per hour. This means the integral is measured in miles. So
is the distance Bob drove from time t = 0 hours to time t = 2 hours. Since we're told 3pm corresponds to t = 0, t = 2 corresponds to 5pm. Putting everything together,
is the distance Bob drove from 3pm to 5pm.
A faucet is turned on and water flows out at a rate of
gallons per minute, where t is the number of minutes since the faucet was turned on. To the nearest gallon, how much water flows out of the faucet during the first two minutes the faucet is turned on?
We want to know the change in the amount of water that has come out of the faucet from time t = 0 minutes to time t = 2 minutes. This means we need to integrate v(t) from 0 to 2.
Rounding to the nearest gallon, 11 gallons come out of the faucet during the first 2 minutes it is turned on.
Air pressure p at h meters above sea level is given by the formula
How much does air pressure change as one moves from 100 meters above sea level to 1000 meters above sea level?
This question is asking for the difference between p(100) and p(1000). We don't even need an integral for this.
As one moves from 100m to 1000m above sea level, the air pressure drops by approximately 10,255 Pascals.
We could use an integral if we wanted. Since we're being asked for the change in pressure between h = 100 and h = 1000, we need to integrate the rate of change of pressure, p'(h), from 100 to 1000. However, using the FTC this gets us the same expression we had when we didn't bother to use the integral:
Fluffles the cat meows more loudly the higher she climbs in her favorite tree. Her volume changes at a rate of
decibels per foot, where h is Fluffles' height above ground measured in feet.
and explain what it represents in the context of this problem.
Since h is given in feet and v(h) in decibels per feet, the integral
is measured in decibels. This integral represents the change in Fluffles' volume as she climbs from 10 feet up in the tree to 20 feet up in the tree. It remains to evaluate the integral.
Fluffles' volume changes by approximately 8 decibels as she climbs from 10 feet up in the tree to 20 feet up in the tree.
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. Can Sonya paint the walls of a 12 foot by 12 foot office with an 8 foot high ceiling in 3 hours?
What about the walls of a 14 foot by 14 foot office with an 8 foot high ceiling?
We need to figure out how many square feet Sonya can paint in 3 hours. Then for each office we need to determine how many square feet there are to paint, and see if this is greater or smaller than the number of square feet Sonya can paint in 3 hours.
To find out how many square feet Sonya can paint in 3 hours, we integrate her rate of painting from time t = 0 hours to time t = 3 hours:
The first office is 12 by 12 feet with a height of 8 feet. Since each wall has width 12 and height 8, and there are 4 walls, the office has
12 × 8 × 4 = 384
square feet of wall that need to be painted. Since 384 is less than 432, Sonya can paint the walls of this office in 3 hours.
The second office is 14 x 14 feet with a height of 8 feet. Since each wall has width 14 and height 8, and there are 4 walls, the office has
14 × 8 × 4 = 448
square feet of wall that need to be painted. Since 448 is greater than 432, Sonya cannot paint the walls of this office in 3 hours.
Gougem Airlines charges customers for checked bags based on weight. Checked bags cost $0.50 per pound up to 50 pounds, and $1 per pound for each additional pound. Let c(p) be the cost per pound for a checked bag, where p is the weight of the bag in pounds. Evaluate
and explain what it means in terms of Gougem Airlines.
The function c(p) is the rate of change of cost with respect to weight. This means
is the change in cost from p = 20 to p = 50, or the change in cost as the bag goes from 20 to 50 pounds. For p ≤ 50 we have c(p) = 0.5, so
As a bag's weight increases from 20 pounds to 50 pounds, the cost to check it increases by
A ball is thrown at the ground from the top of a tall building. The speed of the ball in meters per second is
v(t) = 9.8t + v0,
where t denotes the number of seconds since the ball has been thrown and v0 is the initial speed of the ball (also in meters per second). If the ball travels 25 meters during the first 2 seconds after it is thrown, what was the initial speed of the ball?
The distance the ball travels during the first 2 seconds after it is thrown is given by
Since we know this integral must be equal to 25, we need to solve the equation
We conclude that the ball's initial speed was 2.7 meters per second.
As one dives deeper, the water pressure p in pcf (pounds per cubic foot) at a depth of h feet below the surface of the water changes at a rate of
p'(h) = 62.4 pcf per foot.
If one starts at a depth of 15 feet below the surface of the water, how many feet deeper must one dive for the water pressure to increase by 800 pcf? Round your answer to the nearest foot.
We want to find a depth H in feet such that the change in p from h = 15 to h = H is 800 pcf.
In symbols, we want to find H for which
Rounding to the nearest foot, if one dives to a depth of 28 feet (13 feet deeper since we're already at a depth of 15 deep), the water pressure will increase by 800 pcf.
Shauna starts painting at noon. She can paint (140 – kt) square feet per hour, where t is the number of hours since she started painting and k is a constant accounting for the fact that Shauna slows down as she gets tired. If Shauna paints 100 square feet between 2pm and 3pm, what is k?
To find the number of square feet Shauna paints between 2 and 3pm, we integrate her rate of painting from t = 2 to t = 3:
We see Shauna painted square feet between 2pm and 3pm. Since the problem says she painted 100 square feet during this time interval, we must have
Tom Sawyer is painting a fence at a rate of (200 – 4x) square feet per hour, where x is the number of hours since he started painting. If the fence is 800 square feet, how long will it take him to finish painting the fence? Round your answer to the nearest minute.
We want to find how many hours H it takes Tom to paint 800 square feet. In H hours Tom can paint
square feet, so we want to find H such that
Here we go:
Rearranging, we get
2H2 – 200H + 800 = 0
and dividing through by 2,
H2 – 100H + 400 = 0.
It's not obvious whether we can factor this, but we can use the quadratic formula to solve for H:
The two answers are
Since 95 hours isn't a realistic amount of time to spend painting a fence, we'll throw that answer out and keep the answer that's approximately 4 hours. The amount 0.174 hours is approximately
0.174 × 60 = 10.44 minutes,
so rounding to the nearest minute Tom spent about 4 hours and 10 minutes painting the fence.