if a snowball melts so that its surface area decreases at a rate of 1 cm^2/min, find the rate at which the diameter decreases when the diameter is 43cm. Leave pi in your answer please.
To find the rate at which the diameter decreases, we need to relate the change in surface area to the change in diameter using the formula for the surface area of a sphere.
The surface area formula for a sphere is:
A = 4πr^2,
where A is the surface area and r is the radius.
Since the diameter is twice the radius, we have:
D = 2r, where D is the diameter.
Differentiating both sides with respect to time (t), we get:
dD/dt = 2(dr/dt).
To find the rate at which the diameter decreases, we need to find dr/dt when the diameter is 43 cm.
Given that the surface area decreases at a rate of 1 cm^2/min, we have:
dA/dt = -1 cm^2/min.
Substituting this value and rearranging the formula for surface area, we have:
1 = dA/dt = d(4πr^2)/dt = 8πr(dr/dt).
Simplifying the equation, we get:
dr/dt = 1 / (8πr).
We are given that the diameter is 43 cm, so the radius is half of that:
r = 43/2 = 21.5 cm.
Now we substitute this value into the equation for dr/dt:
dr/dt = 1 / (8π(21.5))
= 1 / (8π * 21.5)
= 1 / (171.2π)
≈ 0.000183 cm/min.
Therefore, the rate at which the diameter decreases when the diameter is 43 cm is approximately 0.000183 cm/min.
To find the rate at which the diameter decreases, we can use the concept of related rates and the formula for the surface area of a sphere. Here's how:
1. Let's denote the diameter of the snowball as D and the rate at which the diameter decreases as dD/dt (the derivative of D with respect to time).
2. We know that the surface area of a sphere is given by the formula A = 4πr^2, where r is the radius. Given that the diameter is D, the radius is half of the diameter, so r = D/2.
3. We are given that the surface area decreases at a rate of 1 cm^2/min. Therefore, dA/dt = -1 cm^2/min (negative because the area is decreasing).
4. We need to find dD/dt, the rate at which the diameter decreases. We can use the chain rule to relate dA/dt and dD/dt:
dA/dt = dA/dr * dr/dD * dD/dt
5. Differentiating the surface area formula with respect to the radius:
dA/dr = 8πr
6. Differentiating the radius (r = D/2) with respect to the diameter:
dr/dD = 1/2
7. Plugging in the values into the chain rule equation:
-1 = 8π(D/2)(1/2)(dD/dt)
8. Simplifying the equation:
-1 = 2πD(dD/dt)
9. Now, we solve for dD/dt:
dD/dt = -1/(2πD)
10. Plugging in the diameter value given (D = 43 cm):
dD/dt = -1/(2π * 43)
= -1/(86π)
Therefore, the rate at which the diameter decreases when the diameter is 43 cm is approximately -1/(86π) cm/min.