Applications of Differential Calculus — Quiz 4

Questions in this quiz (10)

1. If one leg of a right triangle increases at $$4$$ cm/sec while the other leg decreases at $$2$$ cm/sec, find how fast the hypotenuse is changing when the legs are $$5$$ cm and $$12$$ cm.

   • $$\frac{8}{13}$$
   • $$-\frac{4}{13}$$
   • $$\frac{12}{13}$$
   • $$\frac{20}{13}$$

2. The diameter and height of a conical container are both $$6$$ inches, and water leaks out at the rate of $$\frac34$$ cubic inches per second. Find the rate at which the water level is dropping when the diameter of the surface is $$3$$ inches.

   • $$-\frac16$$
   • $$-\frac14$$
   • $$-\frac13$$
   • $$-\frac12$$

3. If the side $$s$$ of a square is increased by $$2\%$$, then the area is increased approximately

   • $$0.02s^2$$
   • $$0.04s^2$$
   • $$0.02s$$
   • $$0.01s^2$$

4. Given $$R=(4\cos(\frac{\pi}{2}t),3\sin(\frac{\pi}{2}t))$$ find a single equation in $$x$$ and $$y$$ for the path.

   • $$16x^2+9y^2=144$$
   • $$\frac{x^2}{16}+\frac{y^2}{9}=1$$
   • $$x^2+y^2=25$$
   • $$9x^2+16y^2=144$$

5. The edge of a cube has length $$8$$ in with a possible error of $$2\%$$. The possible error in the volume is

   • $$5.12$$
   • $$7.68$$
   • $$10.24$$
   • $$30.72$$

6. At the point where $$t=\frac13$$ find the slope of the curve $$R=(2\cos(\pi t),\sin(\pi t))$$

   • $$-\sqrt3$$
   • $$-\frac{\sqrt3}{2}$$
   • $$\sqrt3$$
   • $$\frac{\sqrt3}{2}$$

7. A line with negative slope is drawn through the point $$(2,1)$$ forming a right triangle with the positive axes. The slope of the line forming the triangle of least area is

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

8. The point(s) on the curve $$x^2-y^2=9$$ closest to the point $$(5,0)$$ is (are)

   • $$(3,0)$$
   • $$(\sqrt{10},\pm1)$$
   • $$(4,\pm\sqrt7)$$
   • $$(\sqrt{13},\pm2)$$

9. The first-quadrant point on the curve $$xy^2=12$$ closest to the point $$(1,0)$$ is

   • $$(3,2)$$
   • $$(4,\sqrt3)$$
   • $$(2,\sqrt6)$$
   • $$(1,2\sqrt3)$$

10. An equation of the tangent to the curve with parametric equations $$x=3t-1,\ y=2-t^2$$ at the point where $$t=1$$ is

    • $$x+2y=5$$
    • $$2x+y=3$$
    • $$x-y=2$$
    • $$2x+3y=7$$