If a snowball melts so that its surface area decreases at a rate of 3 cm2/min, find the rate at which the diameter decreases when the diameter is 9 cm. (Give your answer correct to 4 decimal places.)
SA = 4πr^2
d(SA)/dt = 8πr dr/dt
when r = 4.5
3 = 8π(4.5) dr/dt
dr/dt = 3/(36π) cm/min
d(diameter)/dt = 2(3/(36π)) cm/min
= 1/(6π) cm/min or appr. 0.0531 cm/min
45.3434
Why did the snowball go to therapy? It had a meltdown!
To find the rate at which the diameter decreases, we first need to relate the surface area and the diameter of the snowball. The surface area of a sphere is given by the formula A = 4πr², where r is the radius (which is half the diameter).
Given that the surface area decreases at a rate of 3 cm²/min, we can differentiate the formula with respect to time:
dA/dt = 8πr (dr/dt)
We are interested in finding the rate at which the diameter changes, so we can write dr/dt in terms of dd/dt (the rate of change of the diameter) using the relationship dr/dt = (1/2) * dd/dt:
dA/dt = 8πr (1/2) * dd/dt
Now we can plug in the values. When the diameter is 9 cm, the radius is 9/2 cm:
3 cm²/min = 8π(9/2) * (1/2) * dd/dt
Simplifying the equation:
3 = 36π * (1/4) * dd/dt
3 = 9π * dd/dt
dd/dt = 3/9π
Calculating the value:
dd/dt ≈ 0.1061 cm/min
So, the rate at which the diameter decreases when it is 9 cm is approximately 0.1061 cm/min. Keep an eye on those snowballs, they can really shrink fast!
To find the rate at which the diameter decreases, we need to relate the surface area to the diameter of the snowball.
The surface area of a sphere is given by the formula:
A = 4πr^2,
where A is the surface area and r is the radius of the sphere. Since we are given the diameter, which is twice the radius, we can rewrite the formula as:
A = πd^2/2,
where d is the diameter of the sphere.
We are given the rate at which the surface area decreases: dA/dt = -3 cm^2/min.
To find the rate at which the diameter decreases, we differentiate both sides of the equation with respect to time t:
dA/dt = (d(πd^2/2))/dt.
Using the power rule of differentiation, we can simplify the equation:
-3 = π(2d)(dd/dt)/2,
-3 = πd(dd/dt),
dd/dt = -3/πd.
Now we can substitute the given diameter value of 9 cm into the equation to find the rate at which the diameter decreases:
dd/dt = -3/π(9).
Calculating the value gives:
dd/dt ≈ -0.1061 cm/min.
Therefore, when the diameter is 9 cm, the rate at which the diameter decreases is approximately -0.1061 cm/min.