Answer:
The diameter is decreasing at the rate of 0.0434 cm/min.
Step-by-step explanation:
A snowball is spherical, so it's area is given by the following formula:
[tex]A = 4\pi r^{2}[/tex]
The radius is half the diameter, so:
[tex]A = 4\pi (\frac{d}{2})^{2}[/tex]
[tex]A = 4\pi (\frac{d}{2})^{2}[/tex]
[tex]A = \pi d^{2}[/tex]
If a snowball melts so that its surface area decreases at a rate of 3 cm^2/min, find the rate at which the diameter decreases when the diameter is 11 cm.
This is [tex]\frac{dd}{dt}[/tex] when [tex]\frac{dA}{dt} = -3, d = 11[/tex]
[tex]A = \pi d^{2}[/tex]
Applying implicit differentiation:
We have to variables(A and d), so:
[tex]\frac{dA}{dt} = 2\pi d \frac{dd}{dt}[/tex]
[tex]-3 = 22\pi \frac{dd}{dt}[/tex]
[tex]\frac{dd}{dt} = -\frac{3}{22\pi}[/tex]
[tex]\frac{dd}{dt} = -0.0434[/tex]
The diameter is decreasing at the rate of 0.0434 cm/min.
