Sujet: Wheat/cone
Nom: rachel
Qui êtes-vous? étudiant

wheat is poured through a chute at the rate of 10 cubic feet per minute and falls in a conical pile whose bottom radius is always half the altitude. how fast will the circumference of the base be increasing when the pile is 8 feet high?

Hi Rachel,

The volume of a conical pile is changing as wheat is being pored from the chute so you are probably going to need an expression for the volume of a cone. The volume of a cone is given by V = 1/3 π r² h where r is the radius of the base and h is the height.

In your situation you know that the radius is half the height so the volume expression can be changed to

V = 2/3 π r³

You are asked how fast the circumference is changing so you are going to need an expression for the circumference of a circle. The circumference of a circle is given by

C = 2 π r

These are not static equations, they are dynamic. As you pour wheat onto the pile V changes so r changes which forces C to change. In other words V, r and C are all functions of time t.

Differentiate both sides of both equations with respect to t. (Do it and then read on.) The first derivative equation has dV/dt on the left side and the variables on the right side are r and dr/dt. The second derivative equation has dC/dt on the left and the only variable on the right side is dr/dt.

You are asked for dC/dt at a particular time, so from the second derivative equation, if you know dr/dt at this time you can find dC/dt. From the first derivative equation you know dV/dt = 10 cu ft per min and hence if you know r at the particular time of interest you can find dr/dt at this time. What is r at the time when h is 8 ft?

Penny