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

   ans
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.)

      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
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

   *** QuickLaTeX cannot compile formula:
   
   10000e^{0.0277t} <!--%% Where did 0.0277 come from?-->
   
   *** Error message:
   Missing $ inserted.
   leading text: \end{document}
   
   

   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
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?
      \(t = 25\)

      ans
   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

      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)?
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.
      \(G'(t)\)

      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
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.
      \(\int C(t)dt\)

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

       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
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

    *** QuickLaTeX cannot compile formula:
    5%
    
    *** Error message:
    Missing $ inserted.
    leading text: \end{document}
    
    

    efficiency, meaning it can use or store about

    *** QuickLaTeX cannot compile formula:
    5%
    
    *** Error message:
    Missing $ inserted.
    leading text: \end{document}
    
    

    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.