Question

Gavin is designing a water tank in the shape of a rectangular prism. Based on the design, the tank’s length should be 7 feet less than its height and its width should be 8 feet less than its height. Gavin wants the tank to hold a minimum of 840 cubic feet of water. Graph the solution to the inequality that Gavin can use to find the possible heights he should consider for the tank.

Answer 1

Answer:

We have a rectangular prism with dimensions, say,

Length = L, Width = W and Height = H.

It is given that, 'the length of the tank should be 7 feet less than height'

i.e. we get, L <  H - 7.

Also, 'the width of the tank should be 8 feet less than height'.

This gives, W < H - 8.

Since, the minimum volume of the rectangular tank should be 840 cubic feet i.e. Volume ≥ 840.

As , Volume of rectangular prism = L × W × H

Thus, we get,

$L \times W \times H \geq 840$

i.e. $(H-7) \times (H-8) \times H \geq 840$

i.e. $(H^2-8H-7H+56) \times H \geq 840$

i.e. $(H^2-15H+56) \times H \geq 840$

i.e. $H^3-15H^2+56H \geq 840$

i.e. $H^3-15H^2+56H-840\geq 0$

So, the graph corresponding to the inequality is given below.

From the graph we see that, the minimum possible height for the tank is 15 feet.