The volume V of a growing spherical cell is
V =4/3πr^3,
where the radius is measured in micrometers (1 µm = 10^−6m). Find the average rate of change of V with respect to r when r changes from 3 to each of the following.
i)3 to 6µm
ii)3 to 4µm
well, you have the function. The average rate of change is just (change in volume)/(change in radius). So,
(i) (V(6)-V(3))/(6-3)
(ii) (V(4)-V(3))/(4-3)
To find the average rate of change of V with respect to r, we need to calculate the difference in V values divided by the difference in r values.
i) For r changing from 3µm to 6µm:
Let's calculate the V values at the two points:
V(3) = (4/3)π(3^3) = 36π µm^3
V(6) = (4/3)π(6^3) = 288π µm^3
Now, let's find the difference in V:
ΔV = V(6) - V(3) = 288π µm^3 - 36π µm^3 = 252π µm^3
And, calculate the difference in r:
Δr = 6µm - 3µm = 3µm
The average rate of change of V with respect to r is:
average rate of change = ΔV / Δr = (252π µm^3) / (3µm) = 84π µm^2/m
ii) For r changing from 3µm to 4µm:
Let's calculate the V values at the two points:
V(3) = 36π µm^3
V(4) = (4/3)π(4^3) = 256π µm^3
Now, let's find the difference in V:
ΔV = V(4) - V(3) = 256π µm^3 - 36π µm^3 = 220π µm^3
And, calculate the difference in r:
Δr = 4µm - 3µm = 1µm
The average rate of change of V with respect to r is:
average rate of change = ΔV / Δr = (220π µm^3) / (1µm) = 220π µm^2/m
To find the average rate of change of V with respect to r, we need to calculate the change in V divided by the change in r.
i) r changes from 3µm to 6µm:
First, let's calculate V when r first changes from 3µm to 6µm.
V1 = (4/3)π(3^3) = 36π µm^3 (volume when r = 3µm)
Now, let's calculate V when r changes to 6µm.
V2 = (4/3)π(6^3) = 288π µm^3 (volume when r = 6µm)
The change in V is V2 - V1 = 288π - 36π = 252π µm^3.
The change in r is 6µm - 3µm = 3µm.
Therefore, the average rate of change of V with respect to r when r changes from 3 to 6µm is:
(252π µm^3) / (3µm) = 84π µm^2.
ii) r changes from 3µm to 4µm:
First, let's calculate V when r first changes from 3µm to 4µm.
V1 = (4/3)π(3^3) = 36π µm^3 (volume when r = 3µm)
Now, let's calculate V when r changes to 4µm.
V2 = (4/3)π(4^3) = 64π µm^3 (volume when r = 4µm)
The change in V is V2 - V1 = 64π - 36π = 28π µm^3.
The change in r is 4µm - 3µm = 1µm.
Therefore, the average rate of change of V with respect to r when r changes from 3 to 4µm is:
(28π µm^3) / (1µm) = 28π µm^2.
In summary:
i) The average rate of change of V with respect to r when r changes from 3 to 6µm is 84π µm^2.
ii) The average rate of change of V with respect to r when r changes from 3 to 4µm is 28π µm^2.