Q: If the population of a town grows at a rate of $$2.4\%$$ per year, starting at $$90,000$$, in what year will it reach $$120,000$$?

$$\text{Year} = 2016 + \frac{\ln(\frac{120000}{90000})}{\ln(1.024)}$$

Q: If a machine depreciates at $$6.8\%$$ per year, what is its value after $$10$$ years if it was purchased for $$2.5$$ million?

$$2,500,000 \times (0.932$$)^{10}$$

Question 1

Which integral correctly represents the arc length of the Golden Gate Bridge cable modeled by $$y = 0.00037x^2$ for $$|x| \leq 640$$?

Accepted Answer

$$\int_{-640}^{640} \sqrt{1 + (0.00074x)^2$} \, dx$$

Question 2

A sphere of radius $$r is $$sliced by two horizontal planes $$h$$ units apart. What is the formula for the surface area of the resulting spherical zone?

Accepted Answer

$$2\pi r h$$

Question 3

Find the spring constant $$k if a $$force of $$30 N is $$required to stretch a spring $$0.2 m $$from its equilibrium position, assuming Hooke's law.

Accepted Answer

$$k = 150 \, 	ext{N/m}$$

Question 4

What is the work required to compress the spring $$0.4 m $$from its equilibrium position if $$k = 150 \, 	ext{N/m}$$?

Accepted Answer

$$12 \, 	ext{J}$$

Question 5

Which integral represents the work required to pump all water to the top of an inverted conical tank of height $$6 m $$and base radius $$1.5 m$$?

Accepted Answer

$$\$$int_0$$^6 \pi \left(\frac{y}{4}\$$right)^2$$ (6-y) \, dy$$

Question 6

If the population of a town grows at a rate of $$2.4\%$$ per year, starting at $$90,000$$, in what year will it reach $$120,000$$?

Accepted Answer

$$	ext{Year} = 2016 + \frac{\ln(\frac{120000}{90000})}{\ln(1.024)}$$

Question 7

A savings account with an APY of $$3.1\%$$ has an initial deposit of $$1500. $$How long will it take for the balance to reach $$2500$$?

Accepted Answer

$$t = \frac{\ln(\frac{2500}{1500})}{\ln(1.031)}$$

Question 8

The number of cells in a tumor doubles every $$6$$ weeks, starting with $$8$$ cells. After how many weeks will there be $$1500$$ cells?

Accepted Answer

$$t = 6 \cdot \frac{\ln(\frac{1500}{8})}{\ln(2)}$$

Question 9

If a machine depreciates at $$6.8\%$$ per year, what is its value after $$10$$ years if it was purchased for $$2.5$$ million?

Accepted Answer

$$2,500,000 	imes (0.932$$)^{10}$$

Question 10

A water trough has a semicircular cross section of radius $$0.25 m $$and length $$3 m. $$Which integral represents the work required to pump all water out when full?

Accepted Answer

$$\$$int_0$$^{0.25} 3 \cdot 2 \sqrt{0.25^2$ - y^2$} \cdot (0.25 - y) \, dy$$

Question 11

If the length of the water trough is doubled, what happens to the work required to pump the water out?

Accepted Answer

The work required is doubled.

Question 12

If the radius of the water trough is doubled, what happens to the work required to pump the water out?

Accepted Answer

The work required increases by a factor of four.

Question 13

Which formula gives the arc length of the curve $$y = x$$^{3/2}$$ + 2 on$$ $$[1,2]$$?

Accepted Answer

$$\$$int_1$$^2 \sqrt{1 + \left(\frac{3}{2}x$$^{1/2}$$\$$right)^2$$} \, dx$$

Question 14

Which integral represents the surface area generated when $$y = 3x + 4$$ for $$0 \leq x \leq 6 is $$revolved about the x-axis?

Accepted Answer

$$\$$int_0$$^6 2\pi (3x + 4) \sqrt{1 + (3)^2$} \, dx$$

Question 15

Which formula gives the mass of a thin bar with density $$\rho(x) = 1 + \sin x$$ for $$0 \leq x \leq \pi$$?

Accepted Answer

$$\$$int_0$$^{\pi} (1 + \sin x) \, dx$$

Question 16

Which formula gives the derivative of $$f(x) = x \ln x$$?

Accepted Answer

$$\ln x + 1$$

Question 17

Which formula gives the integral $$\int 2x e$$^{x^2}$$ \, dx$$?

Accepted Answer

$$e^{x^2} + C$

Question 18

If $$85\% of $$the original Uranium-238 remains in a rock, and its half-life is $$4.5$$ billion years, how old is the rock?

Accepted Answer

$$t = \frac{\ln(0.85)}{\ln(0.5)} 	imes 4.5 	ext{ billion years}$$

Calculus II Review: Arc Length, Surface Area, Physical Applications, and Exponential Models

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