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

Answer:

6.913 cubic-meters/second.

Step-by-step explanation:

Volume of pyramid is.

$v = \frac{s^2 h}{3}$

and

$\frac{dv}{dt}=35cubic meters/sec.$

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.

$\frac{S}{h} = \frac{6}{8}= \frac{3}{4}$

solving for s and substituting in Volume function gives.

$v =\frac{3h^{3} }{16}$

and taking derivative with respect to time gives.

$\frac{dv}{dt}=\frac{9h^2}{16}\frac{dh}{dt}$

but we have been given that piece of information so.

$35 = \frac{9h^2}{16}\frac{dh}{dt}$

at h = 3 in above we have finally.

$\frac{dh}{dt} = 6.913.$