91Ó°ÊÓ

Volume The radius \(r\) of a sphere is increasing at a rate of 3 inches per minute. (a) Find the rates of change of the volume when \(r=9\) inches and \(r=36\) inches. (b) Explain why the rate of change of the volume of the sphere is not constant even though \(d r / d t\) is constant.

Finding a Derivative IIn Exercises \(9-34,\) find the derivative of the function. \(y=\frac{1}{x-2}\)

Height At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high \(?\) (Hint: The formula for the volume of a cone is \(V=\frac{1}{3} \pi r^{2} h . )\)

Height The volume of oil in a cylindrical container is increasing at a rate of 150 cubic inches per second. The height of the cylinder is approximately ten times the radius. At what rate is the height of the oil changing when the oil is 35 inches high? (Hint: The formula for the volume of a cylinder is \(V=\pi r^{2} h . )\)

Evaporation As a spherical raindrop falls, it reaches a layer of dry air and begins to evaporate at a rate that is proportional to its surface area \(\left(S=4 \pi r^{2}\right) .\) Show that the radius of the raindrop decreases at a constant rate.

Finding the Slope of a Graph In Exercises \(31-38\) , find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. $$f(x)=-\frac{1}{2}+\frac{7}{5} x^{3} \quad\left(0,-\frac{1}{2}\right)$$

Volume Let \(V\) be the volume of a cube of side length \(s\) that is changing with respect to time. If \(d s / d t\) is constant, is \(d V / d t\) constant? Explain.

Flight Control An airplane is flying in still air with an airspeed of 275 miles per hour. The plane is climbing at an angle of \(18^{\circ} .\) Find the rate at which the plane is gaining altitude.

Angle of Elevation A balloon rises at a rate of 4 meters per second from a point on the ground 50 meters from an observer. Find the rate of change of the angle of elevation of the battoon from the observer when the battoon is 50 meters above the ground.

Implicit and Explicit Forms Write two different equations in implicit form that you can write in explicit form. Then write two different equations in implicit form that you cannot write in explicit form.