Use Formulas to Solve Problems
Learning Objectives
- 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 it's side lengths
Example: Solving an Application Using a Formula
It takes Andrew 30 min to drive to work in the morning. He drives home using the same route, but it takes 10 min longer, and he averages 10 mi/h less than in the morning. How far does Andrew drive to work?Answer: This is a distance problem, so we can use the formula [latex]d=rt[/latex], where distance equals rate multiplied by time. Note that when rate is given in mi/h, time must be expressed in hours. Consistent units of measurement are key to obtaining a correct solution. First, we identify the known and unknown quantities. Andrew’s morning drive to work takes 30 min, or [latex]\frac{1}{2}[/latex] h at rate [latex]r[/latex]. His drive home takes 40 min, or [latex]\frac{2}{3}[/latex] h, and his speed averages 10 mi/h less than the morning drive. Both trips cover distance [latex]d[/latex]. A table, such as the one below, is often helpful for keeping track of information in these types of problems.
[latex]d[/latex] | [latex]r[/latex] | [latex]t[/latex] | |
---|---|---|---|
To Work | [latex]d[/latex] | [latex]r[/latex] | [latex]\frac{1}{2}[/latex] |
To Home | [latex]d[/latex] | [latex]r - 10[/latex] | [latex]\frac{2}{3}[/latex] |
Analysis of the Solution
Note that we could have cleared the fractions in the equation by multiplying both sides of the equation by the LCD to solve for [latex]r[/latex].Try It 3
On Saturday morning, it took Jennifer 3.6 h to drive to her mother’s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer 4 h to return home. Her speed was 5 mi/h slower on Sunday than on Saturday. What was her speed on Sunday?Answer: 45 [latex]\frac{text{mi}}{\text{h}}[/latex]
Example: Solving an Area Problem
The perimeter of a tablet of graph paper is 48 in2. The length is [latex]6[/latex] in. more than the width. Find the area of the graph paper.Answer: The standard formula for area is [latex]A=LW[/latex]; however, we will solve the problem using the perimeter formula. The reason we use the perimeter formula is because we know enough information about the perimeter that the formula will allow us to solve for one of the unknowns. As both perimeter and area use length and width as dimensions, they are often used together to solve a problem such as this one. We know that the length is 6 in. more than the width, so we can write length as [latex]L=W+6[/latex]. Substitute the value of the perimeter and the expression for length into the perimeter formula and find the length.
Try It 5
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?Answer: 250 ft2
Example: Solving a Volume Problem
Find the dimensions of a shipping box given that the length is twice the width, the height is [latex]8[/latex] inches, and the volume is 1,600 in.3.Answer: The formula for the volume of a box is given as [latex]V=LWH[/latex], the product of length, width, and height. We are given that [latex]L=2W[/latex], and [latex]H=8[/latex]. The volume is [latex]1,600[/latex] cubic inches.
Analysis of the Solution
Note that the square root of [latex]{W}^{2}[/latex] would result in a positive and a negative value. However, because we are describing width, we can use only the positive result.Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
- Question ID 52436. Authored by: Edward Wicks. License: CC BY: Attribution. License terms: IMathAS Community License CC- BY + GPL.
- Question ID 7679. Authored by: Tyler Wallace. License: CC BY: Attribution. License terms: IMathAS Community License CC- BY + GPL.
- Question ID 1688. Authored by: WebWork-Rochester. License: CC BY: Attribution. License terms: IMathAS Community License CC- BY + GPL.