Question

An inverted pyramid is being filled with water at a constant rate of 35 cubic centimeters per second. The pyramid, at the top, has the shape of a square with sides of length 6 cm, and the height is 8 cm. Find the rate at which the water level is rising when the water level is 3 cm.

Answer 1

6.913 cubic-meters/second.

Explanation:

Volume of pyramid is.

`$v = (s^2 h)/(3) $$`

and

`(dv)/(dt)=35cubic meters/sec.`

we essentially need to compute derivative at h = 3.

but firs we need to write s in terms of h only, to do that we use the fact that ration of side to height of a pyramid is always constant, which means.

`$(S)/(h) = (6)/(8)= (3)/(4) $`

solving for s and substituting in Volume function gives.

`$v =(3h^(3) )/(16) $`

and taking derivative with respect to time gives.

`$(dv)/(dt)=(9h^2)/(16)(dh)/(dt)`

but we have been given that piece of information so.

`$35 = (9h^2)/(16)(dh)/(dt) $`

at h = 3 in above we have finally.

`(dh)/(dt) = 6.913.`