7.6: Homework- Integral Applications

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A car’s velocity follows the equation $v(t) = 10t - t^2$ feet per second from $t = 0$ to $t = 10$. How far does the car travel during this time period?
$\approx 166.7$ feet
ans
A car’s velocity follows the equation $v(t) = 10 - \sqrt{t}$ from $t = 0$ to $t = 100$. How far does the car travel from $t = 0$ to $t = 100$?
$\approx 333.33$ units
ans
A car’s acceleration follows the equation $a(t) = t$ from $t = 0$ to $t = 10$. Recall that acceleration is the derivative of velocity.
1. Find a function $v(t)$ for the velocity at time $t$.
  $v(t) =\frac{1}{2} t^2$ (you could also add any constant to this and still have a valid answer.)
  ans
2. How far does the car travel from $t = 0$ to $t = 10$?
  Need to compute $\int_{0^10} \frac{1}{2} t^2dt \approx 166.7$ units.
  ans
An employee’s wages start at $10,000 a year and quickly increase after that at a rate of $0.04$ per year, continuously implemented. Thus, at year $t$, the employee makes
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dollars per year.
1. How much does the employee make per year at year $5$?
  $\$11485.5$
  ans
2. How much total does the employee make in the first five years?
  $\int_0^5 10000e^{0.0277t}dt \approx \$53628$
  ans
Water drains from a tub at a rate of $\sqrt{50 - 2t}$ gallons per minute, with $t$ measured in minutes.
1. How long does it take for the rate to drop to zero?
  $t = 25$
  ans
2. How much total water has been lost at this point?
  $\int_0^{25} \sqrt{50-2t} dt \approx 69.28$
  ans
A biologist models elk growth rate as $G(t) = 5 e^{0.02 t}$ measured in elk per year.
1. How fast is the elk growth rate changing at $t = 10$?
  $0.122$ elk per year per year
  ans
2. How many elk were born in the first 20 years of this model?
  $\int_0^{20} 5 e^{0.02 t} \approx 123$
  ans
3. Do a sensitivity analysis. Given a small change in $5$, how does that affect the answer to part (a)? Given a small change in $0.02$, how does that affect part (a)?
Let $G(t)$ be the rate at which GDP is growing measured in dollars per day. Match the symbols $G'(t)$, $G(t)$ and $\int_0^t G(t)dt$ to the following statements.
1. This measures the rate that GDP growth is speeding up or slowing down.
  $G'(t)$
  ans
2. This measures how much GDP has increased since the beginning of the year.
  $\int_0^t G(t)dt$
  ans
3. This measure how quickly GDP is increasing.
  $G(t)$
  ans
The Greenland ice sheet is losing ice. It is estimated that it is losing ice at a rate of $f(t) = -0.5t^2-150$ gigatonnes per year, with $t$ measured in years, and $t = 0$ representing $2010$. How many gigatonnes of ice will the ice sheet lose from $2015$ to $2025$?
$\int_5^{15} -0.5t^2-150dt \approx -2040$ gigatonnes.
ans
Let $C(t)$ be the crime rate in the city of Gotham, with $C(t)$ measured in crimes per day, and $t$ measured in days. Match $C(t)$, $C'(t)$, and $\int C(t) dt$ to the following.
1. This function would tell you how many crimes are committed over the last 90 days.
  $\int C(t)dt$
  ans
2. This function would tell you how many crimes per day were being committed 90 days ago.
  $C(t)$
  ans
3. This function will tell you how quickly the crime rate was increasing or decreasing 90 days ago.
  $C'(t)$
  ans
When blasting off from the earth into space, a rocket uses fuel at a rate of $f(t) = 5 + 100e^{-0.01t}$, where $t$ is measured in seconds and $f(t)$ is measured in gallons per second.
1. How many gallons are used in a four-minute flight starting at $t = 0$.
  $\int_0^4 5 + 100e^{-.01t} \approx 412$
  ans
2. How many gallons are used in a two-minute flight starting at $t = 0$?
  $\int_0^2 5 + 100e^{-0.01t} \approx 208$
  ans
3. Should your answer for (b) be exactly half of the answer for part (a)? Why or why not?
  No, since rockets don’t use fuel at a constant rate.
  ans
The amount of sun power that is available to a flower is given by $S(t) = 2.5 \sin\left( \frac{\pi}{12} t \right) + 2.5$ kilojoules per hour. The flower can absorb energy at
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efficiency, meaning it can use or store about
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of the available sunlight energy. How much energy (in kilojoules) does the flower absorb in a $48$-hour period?

$6$ kilojoules
ans
Submarine Navigation
Nuclear submarines spend months underwater with no access to GPS or similar navigation techniques. Instead, they use a “dead reckoning” approach where accelerometers are used to keep track of how fast they are moving, from which their position can be determined. A submarine starts not moving at all. Given the following list of accelerations, estimate how far the submarine has gone.