# 7.6: Homework- Integral Applications

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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
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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.
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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.



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?
$$\int_0^{25} \sqrt{50-2t} dt \approx 69.28$$
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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.
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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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5%

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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
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