Swimming pool question

Question

A swimming pool is 45 ft long and 30 ft wide. The bottom of the pool is an inclined plane which is 3 ft deep at one end and 12 ft deep at the other end. If the pool currently holds 3675 cubic feet of water, what is the approximate maximum depth of water in the pool at the moment?

Answer 1

A swimming pool is 45 ft long and 30 ft wide. The bottom of the pool is an inclined plane which is 3 ft deep at one end and 12 ft deep at the other end. If the pool currently holds 3675 cubic feet of water, what is the approximate maximum depth of water in the pool at the moment?

 

The rectangle portion of pool has volume of

45*30*3 = 4050 cu ft

 

The maximum triangular volume of the pool is

(1/2)* 9*45*30=6075cu ft

 

Since this triangular volume is greater than the volume we have to find, we know the level of water is less, but the volume of water is 3675 cu ft.

 

We set the volume equation to with unknown volumes of X & Y.

(1/2) * 30 * X * Y = 3675

 

We know X & Y, formerly 45 & 9 ft respectively, are a ratio of 45/9 5, therefore we use X/Y = 5 as our second equation.

 

Given two equations and two unknowns, we can solve for Y when the volume in the pool is 3675 cu ft.

 

 

X*Y = 245

 

X/Y = 5    =>  Y = (X/5)

 

X * (X/5) = 245

 

X² = 245 * 5

 

X = 35

 

 Y = (X/5)

 

 Y = 35/5 = 7ft at deep end exactly when volume of pool is 3675 cu ft.

 

I wish I could have included the picture of the pool I sketched