Mickey’s snow cone has completely melted. It is in a cup that has the shape of a right cylindrical cone.

Question

Mickey’s snow cone has completely melted. It is in a cup that has the shape of a right cylindrical cone. This super-sized cup is 8cm across and 14 cm deep. As the snow cone melts, it leaks out of the bottom of the cup at a rate of of 4 cm^3/sec. How fast is the melted snow cone level dropping when the height of the melted snow cone is 3cm?

Answer 1

Given

d = 8cm

r = 4 cm

h = 14 cm

dV/dt = -4 cm^3/s

deltah = 3cm

Find

dh/dt = ?

Solution

V = (1/3) * π * r^2 * h

Please note to simplify this expression we need to develop a relationship between the radius and height

This is a right angle cone, draw the right triangle and apply law of similar right triangle

You have a larger triangle whose adjacent side is 14 cm and opposite side is 4cm

The smaller triangle has an adjacent side of h and opposite side of r

Similar Triangle

r/4 = h/14

r = (2/7) * h

Now sub into volume equation

V = (1/3) * π *((2/7) * h)^2 * h

V = (4/147)*π*h^3

d/dt (V) = (d/dt) *(4/147)*π*h^3

dV/dt = (4/147)*π*(3h^2*dh/dt)

dV/dt = (12/147) *π*h^2*dh/dt

Solve for dh/dt

dh/dt = (147/12π*h^2)*dV/dt

Plug in

h = 14-3=11 cm

dv/dt = -4 cm^3/s

Solve for dh/dt = -0.1289 cm/s