In Exercise, a problem situation is given.(a) Decide what is being asked for, and label th...

In Exercise, a problem situation is given.

(a) Decide what is being asked for, and label the unknown quantities.

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(b) Translate the verbal statements in the problem and the relationships between the known and unknown quantities into mathematical language, using a table as in Examples 1–3. You need not find an equation to be solved.

How many gallons of a 12% salt solution should be combined with 10 gallons of an 18% salt solution to obtain a 16% solution?

EXAMPLE 1

Set up the following problem: The average of two real numbers is 41.125, and their product is 1683. What are the numbers?

SOLUTION

Read: We are asked for two numbers. Label: Call the numbers x and y. Translate:

Consolidate: One technique to use when you have two unknowns is to express one in terms of the other and use this to obtain an equation in one variable. In this case, we can do that by solving the second equation for y:

and substituting the result in the first equation:

The solution of this equation is one of the numbers, and 1683/x is the other.

EXAMPLE 2

Set up the following problem: Arectangle is twice as long as it is wide. If it has an area of 24.5 square inches, what are its dimensions?

SOLUTION

Read: We are asked to find the length and width. Label: Let x denote the width and y the length, and draw a picture of the situation, as in Figure 1. Translate: Use the fact that the area of a rectangle is length × width.

Consolidate: Substitute y = 2x in the area equation:

So the equation to be solved is 2x2 = 24.5.

Figure 1

EXAMPLE 3

Set up this problem: A rectangular box with a square base and no top is to have a volume of 20,000 cubic centimeters. If the surface area of the box is 4000 square centimeters, what are its dimensions?

SOLUTION Read: We must find the length, width, and height of the box. Label: Let x denote the length. Since the base is square, the length and width are the same. Let h denote the height, as in Figure 2. Translate: Recall that the volume of a box is given by the product length × width × height and that the surface area is the sum of the area of the base and the area of the four sides of the box. Then we have these translations:

Consolidate: We have two equations in two variables, so we solve the first equation for h

and substitute this result in the second equation:

The solution of this last equation will provide the solution of the problem.

Figure 2