## Using Formulas to Solve Problems

### Learning Outcomes

• Set up a linear equation involving distance, rate, and time.
• Find the dimensions of a rectangle given the area.
• Find the dimensions of a box given information about its side lengths.

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region, $A=LW$; the perimeter of a rectangle, $P=2L+2W$; and the volume of a rectangular solid, $V=LWH$. When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

### Example: Solving an Application Using a Formula

It takes Andrew 30 minutes to drive to work in the morning. He drives home using the same route, but it takes 10 minutes longer, and he averages 10 mi/h less than in the morning. How far does Andrew drive to work?

### Try It

On Saturday morning, it took Jennifer 3.6 hours to drive to her mother’s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer 4 hours to return home. Her speed was 5 mi/h slower on Sunday than on Saturday. What was her speed on Sunday?

### Example: Solving a Perimeter Problem

The perimeter of a rectangular outdoor patio is $54$ ft. The length is $3$ ft. greater than the width. What are the dimensions of the patio?

### Try It

Find the dimensions of a rectangle given that the perimeter is $110$ cm. and the length is 1 cm. more than twice the width.

### Example: Solving an Area Problem

The perimeter of a tablet of graph paper is 48 in. The length is $6$ in. more than the width. Find the area of the graph paper.

### Try It

A game room has a perimeter of 70 ft. The length is five more than twice the width. How many ft2 of new carpeting should be ordered?

### Example: Solving a Volume Problem

Find the dimensions of a shipping box given that the length is twice the width, the height is $8$ inches, and the volume is 1,600 in.3.

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