Question

Two cylindrical swimming pools are being filled simultaneously at the same rate. The smaller pool has a radius of 5 m, and the water level rises at a rate of 0.5m/min. The larger pool has a radius of 8 m. How fast is the water level rising in the larger pool?

Answer 1

Answer:

0.195m/min

Step-by-step explanation:

The radii of the smaller pool =5m

The radii of the larger pool = 8m

DEFINITION:

Given two similar shapes, and the ratio of lengths in the two similar shapes, the ratio of the areas is a square of the ratio of the lengths.

The radii of the pools have a ratio of 5:8

Therefore, the ratio of surface areas will be  $5^2 : 8^2=25:64$

Since the pools are being filled at the same rate  

Let the Volume of the Smaller Pool, $V_s$ and the height of the water be $H_s$

Let the Volume of the Larger Pool, $V_L$ and the height of the water be $H_L$

Volume of a cylinder = Base surface area X Height

$V_s=25H_s$

$V_L=64H_L$

$\frac{dV_s}{dt} =25\frac{dH_s}{dt} \\\frac{dV_L}{dt} =64\frac{dH_L}{dt} \\\frac{dV_s}{dt}=\frac{dV_L}{dt} \text{ Since the water inflow is the same rate}\\25\frac{dH_s}{dt}=64\frac{dH_L}{dt}\\\frac{dH_s}{dt}=0.5 m/min$

$25X0.5=64\frac{dH_L}{dt}\\12.5=64\frac{dH_L}{dt}\\\frac{dH_L}{dt}=\frac{12.5}{64}=0.195m/min$

The water level is rising in the larger pool at a rate of 0.195m/min