Applications of Differential Calculus — Quiz 6

Questions in this quiz (9)

1. If $$h'(c)$$ satisfies the Mean Value Theorem for $$h(x)=\ln(x+2)$$ on $$[0,b]$$, find the smallest possible value of $$b$$.

   • $$1$$
   • $$2$$
   • $$3$$
   • $$4$$

2. On what intervals is the function $$g(x)=\sin x\cos x$$ concave up on $$[0,2\pi]$$? (A) $$(0,\frac{\pi}{2})\cup(\frac{3\pi}{2},2\pi)$$ (B) $$(\frac{\pi}{2},\frac{3\pi}{2})$$ (C) $$(\frac{\pi}{2},\pi)\cup(\frac{3\pi}{2},2\pi)$$ (D) $$(\pi,2\pi)$$

   • $$(0,\frac{\pi}{2})\cup(\frac{3\pi}{2},2\pi)$$
   • $$(\frac{\pi}{2},\frac{3\pi}{2})$$
   • $$(\frac{\pi}{2},\pi)\cup(\frac{3\pi}{2},2\pi)$$
   • $$(\pi,2\pi)$$

3. Find the maximum area of a rectangle that has two vertices on the x-axis and two vertices on the graph $$y=9-x^2$$. (A) $$12\sqrt3$$ (B) $$24$$ (C) $$36$$ (D) $$54$$

   • $$12\sqrt3$$
   • $$24$$
   • $$36$$
   • $$54$$

4. A rectangular open-top box with height $$3$$ inches must have volume $$72$$ in$$^3$$. The longer sides cost $$200$$ per in$$^2$$, shorter sides cost $$350$$ per in$$^2$$, and the bottom costs $$0.05$$ per in$$^2$$. What dimensions minimize cost? (A) $$6.48\times3.70\times3$$ (B) $$4\times6\times3$$ (C) $$3\times8\times3$$ (D) $$2\times12\times3$$

   • $$6.48\times3.70\times3$$
   • $$4\times6\times3$$
   • $$3\times8\times3$$
   • $$2\times12\times3$$

5. A balloon is expanding in the shape of a sphere. The radius is increasing at a constant rate of $$3$$ ft/sec. At what rate is the surface area increasing when the radius is $$20$$ ft? (A) $$1440\pi$$ (B) $$2880\pi$$ (C) $$480\pi$$ (D) $$960\pi$$

   • $$1440\pi$$
   • $$2880\pi$$
   • $$480\pi$$
   • $$960\pi$$

6. A particle moves along the path defined by $$x=\sin t,\ y=\cos 2t$$. Find the acceleration vector at $$t=\frac{5\pi}{6}$$. (A) $$\langle-\frac12,-2\rangle$$ (B) $$\langle\frac12,-\sqrt3\rangle$$ (C) $$\langle-\sqrt3,-2\rangle$$ (D) $$\langle\frac{\sqrt3}{2},2\rangle$$

   • $$\langle-\frac12,-2\rangle$$
   • $$\langle\frac12,-\sqrt3\rangle$$
   • $$\langle-\sqrt3,-2\rangle$$
   • $$\langle\frac{\sqrt3}{2},2\rangle$$

7. A farmer is building a pen along a straight river. One side is the river so no fencing is needed. The farmer has $$300$$ ft of fencing and must include at least $$80$$ ft along the river. The farmer may build either a rectangular pen or semicircular pen. Which produces the greatest area? (A) Rectangle $$75$$ by $$75$$ (B) Rectangle $$100$$ by $$50$$ (C) Semicircle radius $$75$$ (D) Semicircle radius $$50$$

   • Rectangle $$75$$ by $$75$$
   • Rectangle $$100$$ by $$50$$
   • Semicircle radius $$75$$
   • Semicircle radius $$50$$

8. Two cars move on straight roads that are at a $$50^\circ$$ angle to each other. If car $$A$$ moves at a constant rate of $$40$$ mph, car $$B$$ moves at a constant rate of $$35$$ mph, and they started from the intersection at the same time, what is the rate of change of the distance between the cars once car $$A$$ has traveled $$25$$ miles?

   • $$27.07$$ mph
   • $$30.15$$ mph
   • $$32.04$$ mph
   • $$40.35$$ mph

9. Suppose liquid is flowing into a vessel at a constant rate. The vessel has the shape of a hemisphere capped by a cylinder. Graph $$y=h(t)$$ the height $$\left(=\text{depth}\right)$$ of the liquid at time $$t,$$ labeling and explaining any salient characteristics of the graph.

   • The graph starts slowly, increases with increasing slope (concave up) until height $$a$$, then becomes a straight line with constant slope from $$a$$ to $$b$$.
   • The graph is a straight line from the origin to height $$b$$, indicating constant slope throughout.
   • The graph starts as a straight line from the origin to height $$a$$, then becomes concave down from $$a$$ to $$b$$.
   • The graph starts at $$\left(0,0\right),$$ is increasing and concave down while the vessel is filling the hemispherical part, passes through the transition height $$y=a,$$ and then becomes a straight line with constant positive slope while the cylindrical part fills, continuing up to the top height $$y=b.$$