Chapter 5.5, Problem 33E

To determine

To find: the number of months the reservoir contains the same amount of water.

Answer to Problem 33E

The amount of water in reservoir $A$ and $B$ will be equal after about $13.33$ months.

Explanation of Solution

Given:

  $34$ million gallons of water in reservoir $A$ .

  $38$ million gallons in reservoir $B$ .

During drought:

Reservoir $A$ loses about $0.8$ million gallons.

Reservoir $B$ loses about $1.1$ million gallons.

Concept used:

The intersection of two straight line can be calculated as:

By equating the value of $x\text{ and }y$ which is same in the case of intersection.

Set the two equations equal to each other and solve for the point where it is intersects.

Calculation:

The objective is to find the number of months after which two reservoirs have same amount of water.

Let $x$ be the number of months.

Verbal model can be written as:

Gallons of gallons lost number =Gallons of- gallon lost number of waters in A per months water in $C$ per months.

Equation:

  $34-0.8x=38-1.1x$ .

Step1: write a system of linear equations using each side of the original equation.

Let

  $\begin{array}{l}y=34-0.8x\mathrm{........}\left(i\right)\\ y=38-1.1x\mathrm{...........}\left(ii\right)\end{array}$

Step 2: solved by graphing.

The graph intersects between $x=6\text{ and }x=7.$

$X$	$Y_1$	$Y_2$
$13.1$	$23.52$	$23.59$
$13.2$	$23.44$	$23.48$
$13.3$	$23.36$	$23.37$
$13.4$	$23.28$	$23.26$
$13.5$	$23.2$	$23.15$
$13.6$	$23.12$	$23.04$
$13.8$	$23.04$	$22.93$

$13.31$	$23.35$	$23.369$
$13.32$	$23.34$	$23.348$
$13.33$	$23.33$	$23.327$
$13.34$	$23.32$	$23.306$
$13.35$	$23.31$	$23.285$
$13.36$	$23.3$	$23.264$
$13.37$	$23.29$	$23.243$
$13.38$	$23.28$	$23.222$

It is clear from the graph that both the lines intersect at $x=13.33$ .

The graph intersects between $x=6\text{ and }x=7$ :

  

Hence, the amount of water in reservoir $A$ and $B$ will be equal after about $13.33$ months.