7.6: Homework- Integral Applications

1. 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
2. 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

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3. 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.)

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   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
4. 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 $5$?
      $\$11485.5$

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   2. How much total does the employee make in the first five years?
      $\int_0^5 10000e^{0.0277t}dt \approx \$53628$

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5. 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?
      $t = 25$

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   2. How much total water has been lost at this point?
      $\int_0^{25} \sqrt{50-2t} dt \approx 69.28$

      ans
6. 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

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   2. How many elk were born in the first 20 years of this model?
      $\int_0^{20} 5 e^{0.02 t} \approx 123$

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   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)?
7. 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.
      $G'(t)$

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   2. This measures how much GDP has increased since the beginning of the year.
      $\int_0^t G(t)dt$

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   3. This measure how quickly GDP is increasing.
      $G(t)$

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8. 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
9. 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.
      $\int C(t)dt$

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   2. This function would tell you how many crimes per day were being committed 90 days ago.
      $C(t)$

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   3. This function will tell you how quickly the crime rate was increasing or decreasing 90 days ago.
      $C'(t)$

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10. 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 $t = 0$.
       $\int_0^4 5 + 100e^{-.01t} \approx 412$

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    2. How many gallons are used in a two-minute flight starting at $t = 0$?
       $\int_0^2 5 + 100e^{-0.01t} \approx 208$

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    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.

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11. 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
12. 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.