Question 1

A car moves along a straight road with velocity $$v(t) = 3t^2$ - 12t + 9 (in m/s$$), where $$t is in $$seconds. What is the displacement of the car from $$t = 0 to$$ $$t = 4$$ seconds?

Accepted Answer

$$\displaystyle \$$int_0$$^4 (3t^2$ - 12t + 9)\,dt = -4 m$$

Question 2

Given the velocity function $$v(t) = 2t^2$ - 8t + 6$$ for $$0 \leq t \leq 3$$, what is the total distance traveled by the object from $$t = 0 to$$ $$t = 3$$?

Accepted Answer

$$\displaystyle \$$int_0$$^3 |2t^2$ - 8t + 6|\,dt = \frac{16}{3}$$ miles

Question 3

A skydiver jumps from a helicopter. Her velocity is given by $$v(t) = 80$$ for $$0 \leq t \u003c 19$$, $$v(t) = 783 - 37t$$ for $$19 \leq t \u003c 21$$, and $$v(t) = 6$$ for $$21 \leq t \u003c 40. $$What is the total distance she falls in the first 40 seconds?

Accepted Answer

$$\displaystyle \$$int_0$$^{19} 80\,dt + \int_{19}^{21} (783 - 37t)\,dt + \int_{21}^{40} 6\,dt = 1720 m$$

Question 4

If the acceleration of an object is $$a(t) = -9.8 m/s$^2$$ and its initial velocity is $$v(0) = 300 m/s$$, what is the velocity function $$v(t)$$?

Accepted Answer

$$v(t) = 300 - 9.8t$$

Question 5

A population grows at a rate $$P'(t) = 30(1 + \sqrt{t})$$ people per year, with $$P(0) = 250. $$What is the population after $$t$$ years?

Accepted Answer

$$P(t) = 250 + 30\left(t + \frac{2}{3}t$$^{3/2}$$\right)$$

Question 6

Given $$v(t) = \frac{1}{4}\sin t$$ and $$s(0) = -\frac{1}{4}$$, what is the position function $$s(t)$$?

Accepted Answer

$$s(t) = -\frac{1}{4}\cos t$$

Question 7

A ball is thrown upward from a height of 30 m with an initial velocity of 300 m/s. At what time does it reach its highest point?

Accepted Answer

$$t = \frac{300}{9.8} \approx 30.6 s$$

Question 8

Which integral gives the net change in a quantity $$Q(t)$$ from $$t = a to$$ $$t = b$$?

Accepted Answer

$$\displaystyle \int_a^b Q'(t)\,dt$$

Question 9

If $$v(t) is $$the velocity of an object, which of the following expressions gives the total distance traveled from $$t = a to$$ $$t = b$$?

Accepted Answer

$$\displaystyle \int_a^b |v(t)|\,dt$$

Question 10

A chemical reaction produces a substance at a rate $$R(t) = 5e$$^{-0.2t}$$ grams per hour. How much of the substance is produced in the first 10 hours?

Accepted Answer

$$\displaystyle \$$int_0$$^{10} 5e$$^{-0.2t}$$\,dt = 25(1 - e$$^{-2}$$)$$ grams

Question 11

A reservoir is being filled at a rate $$Q'(t) = 100 - 4t$$ liters per hour. If it starts empty, how much water is in the reservoir after 10 hours?

Accepted Answer

$$Q(10) = \$$int_0$$^{10} (100 - 4t)\,dt = 1000 - 200 = 800$$ liters

Question 12

If $$s(t) is $$the position function and $$v(t) = s'(t)$$, which of the following correctly expresses the position at time $$t$$ given $$s(0)$$?

Accepted Answer

$$s(t) = s(0) + \$$int_0$$^t v(x)\,dx$$

Question 13

A particle moves along a line so that its velocity is $$v(t) = t - 2. $$What is the displacement from $$t = 0 to$$ $$t = 4$$?

Accepted Answer

$$\$$int_0$$^4 (t - 2)\,dt = 0$$

Question 14

A medication is administered at a rate $$r(t) = 20e$$^{-0.1t}$$ mg/hour. What is the total amount administered in the first 5 hours?

Accepted Answer

$$\$$int_0$$^5 20e$$^{-0.1t}$$\,dt = 200(1 - e$$^{-0.5}$$) mg$$

Question 15

If $$Q'(t) is $$the rate of change of a quantity $$Q(t)$$, which formula gives the future value $$Q(t)$$ given $$Q(0)$$?

Accepted Answer

$$Q(t) = Q(0) + \$$int_0$$^t Q'(x)\,dx$$

Chapter 6: Applications of Integration – Velocity, Net Change, and Physical Applications

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