Homework: Integral Applications

Tyler Seacrest

69 Homework: Integral Applications

A car’s velocity follows the equation $v (t) = 10 t - t^{2}$ 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 $\int_{0^{1} 0} \frac{1}{2} t^{2} d t \approx 166.7$ 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
$10000 e^{0.0277 t}$

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} 10000 e^{0.0277 t} d t \approx $ 53628$
  
  ans
Water drains from a tub at a rate of $\sqrt{50 - 2 t}$ 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?
  
  $\int_{0}^{25} \sqrt{50 - 2 t} d t \approx 69.28$
  
  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 $\int_{0}^{t} G (t) d t$ 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.
  
  $\int_{0}^{t} G (t) d t$
  
  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 $f (t) = - 0.5 t^{2} - 150$ 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 $2025$ ?

$\int_{5}^{15} - 0.5 t^{2} - 150 d t \approx - 2040$ 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 $\int C (t) d t$ to the following.
1. This function would tell you how many crimes are committed over the last 90 days.
  
  $\int C (t) d t$
  
  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 $f (t) = 5 + 100 e^{- 0.01 t}$ , 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 $t = 0$ .
  
  $\int_{0}^{4} 5 + 100 e^{- .01 t} \approx 412$
  
  ans
2. How many gallons are used in a two-minute flight starting at $t = 0$ ?
  
  $\int_{0}^{2} 5 + 100 e^{- 0.01 t} \approx 208$
  
  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 $S (t) = 2.5 \sin (\frac{π}{12} t) + 2.5$ kilojoules per hour. The flower can absorb energy at $5$ efficiency, meaning it can use or store about $5$ 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.

$\begin{array}{cc} Day & Average Acceleration in miles / day^{2} \\ 0 & 220 \\ 1 & 135 \\ 2 & - 150 \\ 3 & 0 \\ 4 & 0 \\ 5 & 200 \\ 6 & 0 \\ 7 & - 405 \\ 8 & 0 \end{array}$

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69 Homework: Integral Applications

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