Sarah is filling ice cream cones with shaving cream to hand out to trick or treaters. The cones have a largest radius of 3 centimeters and a depth of 13 centimeters. She fills them at a rate of 5 cubic centimeters per second. (She works carefully so that shaving cream is compact and even) At what rate is the level of the shaving cream changing when it is still 3 centimeters from the top of the cone?
The trick is that the change in volume is the surface area times the change in height.
dV/dt = pi r^2 dh/dt
here dV/dt is constant = 5 cm^3/s
so all you need is pi r^2 at depth of 11 cm
depth of 10 cm I mean :)
Traditional approach,
at a time of t seconds, let the height of cream be h, and the radius of the cream be r
by similar shapes,
h/r = 13/3
3h = 13r ---> h = 13r/3
V = (1/3)π r^2 h
= (1/3)π r^2 (13r/3)
= (13/9)π r^3
dV/dt = 13/3 π r^2 dr/dt
5 = 13/3 π r^2 dr/dt
15/(13πr^2) = dr/dt
when h = 10, r = 30/13
dr/dt = 15/(13π(900/169)
= 13/(60π)
"level of the shaving cream changing" ?? I will interpret that as how fast is the area of the surface changing.
A = πr^2
dA/dt = 2π r dr/dt
so when r = 30/13
dA/dt = 2π(30/13)(13/(60π))
= 1 cm^2 / s
To find the rate at which the level of the shaving cream is changing when it is still 3 centimeters from the top of the cone, we can use related rates.
Let's define some variables:
- Let r be the radius of the shaving cream at a certain point in time.
- Let h be the height of the shaving cream at a certain point in time.
- Let V be the volume of the shaving cream at a certain point in time.
We know that the largest radius of the cone is 3 centimeters. This means that when the level of the shaving cream is still 3 centimeters from the top of the cone, the height of the shaving cream is 13 - 3 = 10 centimeters.
The volume of a cone can be calculated using the formula V = (1/3) * π * r^2 * h. Since the shaving cream is being filled at a rate of 5 cubic centimeters per second, we can express the rate of change of volume as dV/dt = 5 cm^3/s.
We want to find dh/dt, the rate at which the height of the shaving cream is changing when it is still 3 centimeters from the top of the cone.
To find dh/dt, we can use the chain rule from calculus:
dh/dt = (dh/dV) * (dV/dt)
First, let's find dh/dV. We can rearrange the volume formula to solve for h: h = (3V) / (π * r^2)
Taking the derivative of both sides with respect to V, we get:
dh/dV = (3 / (π * r^2))
Now, let's find dV/dt. Since we know dV/dt = 5 cm^3/s, we can substitute that into the chain rule equation.
dh/dt = (dh/dV) * (dV/dt)
dh/dt = (3 / (π * r^2)) * (5 cm^3/s)
To calculate dh/dt, we need to know the radius of the shaving cream at the given point. Since it's not mentioned in the question, we are missing this crucial information. Without the value of r, we cannot calculate the rate at which the level of the shaving cream is changing when it is still 3 centimeters from the top of the cone.