Answer :
Answer:
The diameter decreases at a rate of -0.0477 cm/min when it is 10 cm.
Step-by-step explanation:
Surface area of a snowball:
A snowball has a spheric format, which means that it's surface area is given by:
[tex]A = 4\pi r^2[/tex]
In which r is the radius, which is half the diameter. In function of the diameter, the area is given by:
[tex]A = 4\pi(\frac{d}{2})^2 = \pi d^2[/tex]
Solving the question:
To solve this question, we have to implicitly derivate the area in function of t. So
[tex]\frac{dA}{dt} = 2d\pi\frac{dd}{dt}[/tex]
Snowball melts so that its surface area decreases at a rate of 3 cm2/min
This means that [tex]\frac{dA}{dt} = -3[/tex]
Find the rate at which the diameter decreases when the diameter is 10 cm.
This is [tex]\frac{dd}{dt}[/tex] when [tex]d = 10[/tex]. So
[tex]\frac{dA}{dt} = 2d\pi\frac{dd}{dt}[/tex]
[tex]-3 = 20\pi\frac{dd}{dt}[/tex]
[tex]\frac{dd}{dt} = -\frac{3}{20\pi}[/tex]
[tex]\frac{dd}{dt} = -0.0477[/tex]
The diameter decreases at a rate of -0.0477 cm/min when it is 10 cm.