Exam 29: Applied Applications of the Derivative

A car is on a road traveling due north at $56.3 \mathrm{~km} / \mathrm{h}$ and a motorcycle is traveling on another road due west at $64.37 \mathrm{~km} / \mathrm{h}$ . The car is $24.14 \mathrm{~km}$ from the point where the roads meet and the motorcycle is $8.00 \mathrm{~km}$ past that point. What is the closest that the two vehicles get?

The tent is in the shape of a triangular prism, with both ends covered, but with no bottom. It has a rectangular base with a width that is half its length. The tent must have a volume of $58.0 \mathrm{~m}^{3}$ . Find the width of the tent that minimizes the surface area of the tent.

A point moves according to the equation $s=3.0 t^{3}-64 t$ , where $s$ is in centimetres and $t$ is in seconds. (a) Find the time $t$ in seconds greater than zero when the point comes to rest and the distance it traveled in this time. (b) Find the non-zero time when the point crosses its starting point.

Find the area of the largest rectangle with sides parallel to the coordinate axes which can be inscribed in the figure bounded by the parabolas $2 x=y^{2}-10$ and $4 x=16-2 y^{2}$ .

A boat with an anchor on the bottom at a depth of $30.0 \mathrm{~m}$ is drifting away from the anchor at $3.00 \mathrm{~m} / \mathrm{s}$ , while the anchor cable slips out at water level. At what rate is the cable leaving the boat when the boat has drifted $10.0 \mathrm{~m}$ away from the spot directly above the anchor? Assume that the cable is straight.

The cost of manufacturing a crankshaft per hour follows the equation $c=2 x^{2}-8 x+2$ where $\mathrm{x}$ is the number of crankshafts. What is the optimum number of crankshafts to minimize production costs?

A $3.5-\mathrm{m}$ ladder is leaning against the side of a building. At what rate is the top of the ladder sliding down the building if the bottom of the ladder is $0.80 \mathrm{~m}$ from the base of the building and being pulled away from the building at $2.0 \mathrm{~m} / \mathrm{s}$ ?

The air in a container is at a pressure of $27.1 \mathrm{lb} / \mathrm{in}^{3}$ when its volume is $96.0 \mathrm{in}^{3}$ . Find the rate of change of the volume with respect to pressure as the pressure increases. Use Boyles' Law, $p v=k$ .

What is the minimum amount of metal needed to make a $300000-\mathrm{cm}^{3}$ cylindrical drum? Express our answer to the nearest whole square centimetre.

The current in a $1.00-\mathrm{H}$ inductor is given by $i=\sqrt{6.02 t^{2}+20.1} t$ . Find the voltage across the inductor at $6.00 \mathrm{~s}$ .

The formula for the volume of fluid $V$ in a spherical tank of radius $r$ when the depth of the fluid $h$ is $V=\frac{\pi h^{2}}{3}(3 r-h)$ A $21.7 \mathrm{~m}$ wide tank is filling with liquid oxygen at a constant rate of $20.5 \mathrm{~L} / \mathrm{s}$ . How fast is the surface of the oxygen rising when the height is $1.06 \mathrm{~m}$ ?