Previously asked:
the length of a rectangular prism is increasing at a rate of 8 cm/s, its width is increasing at a rate of 3 cm/s, and its height is increasing at a rate of 5 cm/s. when the length is 20 cm, width is 10 cm, and height is 15 cm, how fast is the volume of the rectangular prism increasing?
I was told: V = lwh
dV/dt = lw dh/dt + lh dw/dt + wh dl/dt
you are given all the values, just plug in and evaluate.
I got 265 as the answer. Is this correct?
Just plug in the numbers you have, and you get
(20)(10)(5) + (20)(15)(3) + (10)(15)(8) = 3100
How did you come up with 265?
To find the rate at which the volume of the rectangular prism is increasing, we need to use the formula for the derivative of the volume with respect to time:
dV/dt = lw dh/dt + lh dw/dt + wh dl/dt
Given that the length is increasing at a rate of 8 cm/s (dl/dt = 8), the width is increasing at a rate of 3 cm/s (dw/dt = 3), and the height is increasing at a rate of 5 cm/s (dh/dt = 5), we can substitute these values into the formula:
dV/dt = (20)(10)(5) + (20)(15)(3) + (10)(15)(8)
= 1000 + 900 + 1200
= 3100 cm^3/s
Therefore, the rate at which the volume of the rectangular prism is increasing is 3100 cm^3/s, not 265.