A sphere has a volume of 36π cubic
meters. The rate of change in volume is
9π cubic meters per minute. Find the
rate of change in area of the sphere
v = 4/3 πr^3
a = 4πr^2
dv/dt = 4πr^2 da/dt = a da/dt
So, find a when v = 36π and plug it in to find da/dt.
oops. forgot about r.
dv/dt = a dr/dt
da/dt = 8πr dr/dt
So, find r and a when 36π and plug them in to find da/dt
Why did the sphere go to therapy? Because it had a lot of issues with its volume.
To find the rate of change in the area of the sphere, we can use the formula for the rate of change of volume with respect to time:
Rate of Change of Volume = 4πr^2 * Rate of Change of Radius
Since we're given the rate of change in volume (9π cubic meters per minute), we can solve for the rate of change in the area.
36π = 4πr^2 * Rate of Change of Radius
Dividing both sides by 4πr^2, we get:
9 = Rate of Change of Radius
So, the rate of change in the area of the sphere is 9 units per minute. Keep rolling with those calculations!
To find the rate of change in the area of the sphere, we need to differentiate the formula for the volume of a sphere with respect to time.
The formula for the volume of a sphere is given by:
V = (4/3)πr^3, where V is the volume and r is the radius of the sphere.
Differentiating both sides of the equation with respect to time, we get:
dV/dt = d/dt[(4/3)πr^3]
Since the rate of change in volume is given as 9π cubic meters per minute, we can substitute this value into the equation:
9π = d/dt[(4/3)πr^3]
Next, we need to solve for dA/dt, the rate of change in the area of the sphere. The formula for the surface area of a sphere is:
A = 4πr^2, where A is the area and r is the radius of the sphere.
Differentiating both sides of the equation with respect to time, we get:
dA/dt = d/dt[4πr^2]
Now, we can relate dA/dt and dV/dt using the given rate of change in volume and the relationship between volume and radius. The volume and radius are related by:
V = (4/3)πr^3
To solve for dr/dt, the rate of change in the radius of the sphere, we need to differentiate both sides of the equation:
dV/dt = d/dt[(4/3)πr^3]
9π = d/dt[(4/3)πr^3]
Now, we can solve for dr/dt by rearranging the equation:
dr/dt = (dV/dt) / [(4/3)πr^2]
Substituting the given values, we have:
dr/dt = (9π) / [(4/3)πr^2]
Simplifying the equation, we get:
dr/dt = (27/4) / r^2
Finally, we can relate dA/dt and dr/dt using the formula for the surface area of a sphere:
dA/dt = d/dt[4πr^2]
Substituting the value of dr/dt into the equation, we have:
dA/dt = d/dt[4πr^2] = 8πr(dr/dt)
Now, we can substitute the value of dr/dt we found earlier:
dA/dt = 8πr[(27/4) / r^2] = (27π/2) / r
Therefore, the rate of change in the area of the sphere is (27π/2) / r.
To find the rate of change in the area of the sphere, we can use the relationship between the volume and the radius of the sphere. The volume of a sphere is given by the formula V = (4/3)πr^3, where V is the volume and r is the radius.
Given that the volume is changing at a rate of 9π cubic meters per minute, we can differentiate the volume formula with respect to time (t) to find the rate of change in volume with respect to time. Thus, we get dV/dt = (4/3)π(3r^2)(dr/dt).
Now, let's substitute the given values into the equation. We know that the volume is 36π cubic meters, so we can set V = 36π. Taking the derivative of the volume formula, we have dV/dt = (4/3)π(3r^2)(dr/dt).
Now we have: dV/dt = 36π = (4/3)π(3r^2)(dr/dt).
Simplifying the equation, we can cancel out the π's and the 3's, which leaves us with: 36 = 4r^2(dr/dt).
Solving for dr/dt, the rate of change in the radius with respect to time, we divide both sides of the equation by 4r^2: (dr/dt) = 36 / (4r^2) = 9 / r^2.
The rate of change in the area of the sphere can be calculated using the formula for the surface area of a sphere, A = 4πr^2. Taking the derivative of the area formula with respect to time, we get dA/dt = 8πr(dr/dt).
Now, plugging in the value we found for dr/dt, we can replace (dr/dt) in the equation with 9 / r^2. So, we have: dA/dt = 8πr * (9 / r^2) = 8π(9 / r).
Therefore, the rate of change in the area of the sphere is given by dA/dt = 72π / r.