Respuesta :
Answer:
0.0221 cm/min
Step-by-step explanation:
Surface area of a snowball:
An snowball has a spherical format.
The surface area of an sphere is given by:
[tex]A = 4\pi r^2[/tex]
In which r is the radius(half the diameter). So in function of the diameter, we have that:
[tex]A = 4\pi (\frac{d}{2})^2 = \pi d^2[/tex]
Implicit derivative:
To solve this question, we need to find the implicit derivative of A in function of t.
The variables are A and d, so:
[tex]\frac{dA}{dt} = 8\pi d \frac{dd}{dt}[/tex]
Its surface area decreases at a rate of 5 cm2/min
This means that [tex]\frac{dA}{dt} = -5[/tex]
Find the rate (in cm/min) at which the diameter decreases when the diameter is 9 cm
This is [tex]\frac{dd}{dt}[/tex] when [tex]d = 9[/tex]. So
[tex]\frac{dA}{dt} = 8\pi d \frac{dd}{dt}[/tex]
[tex]-5 = 8\pi*9 \frac{dd}{dt}[/tex]
[tex]\frac{dd}{dt} = -\frac{5}{72\pi}[/tex]
[tex]\frac{dd}{dt} = -0.0221[/tex]
This means that the diameter decreases at a rate of 0.0221 cm/min when the diameter is 9 cm.
