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Lee wants to drive from Phoenix to his brother’s apartment in San Francisco, a distance of 770 miles. If he drives at a steady rate of 70 miles per hour, how many hours will the trip take?
11 hours
Yesenia is 168 miles from Chicago. If she needs to be in Chicago in 3 hours, at what rate does she need to drive?
56 mph
You are probably familiar with some geometry formulas. A formula is a mathematical description of the relationship between variables. Formulas are also used in the sciences, such as chemistry, physics, and biology. In medicine they are used for calculations for dispensing medicine or determining body mass index. Spreadsheet programs rely on formulas to make calculations. It is important to be familiar with formulas and be able to manipulate them easily.
In [link] and [link] , we used the formula $d=rt$ . This formula gives the value of $d$ , distance, when you substitute in the values of $r\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t$ , the rate and time. But in [link] , we had to find the value of $t$ . We substituted in values of $d\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ and then used algebra to solve for $t$ . If you had to do this often, you might wonder why there is not a formula that gives the value of $t$ when you substitute in the values of $d\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ . We can make a formula like this by solving the formula $d=rt$ for $t$ .
To solve a formula for a specific variable means to isolate that variable on one side of the equals sign with a coefficient of 1. All other variables and constants are on the other side of the equals sign. To see how to solve a formula for a specific variable, we will start with the distance, rate and time formula.
Solve the formula $d=rt$ for $t$ :
We will write the solutions side-by-side to demonstrate that solving a formula in general uses the same steps as when we have numbers to substitute.
ⓐ when $d=520$ and $r=65$ | ⓑ in general | ||||
Write the formula. | $\phantom{\rule{1em}{0ex}}d=rt$ | Write the formula. | $d=rt$ | ||
Substitute. | $520=65t$ | ||||
Divide, to isolate $t$ . | $\frac{520}{65}=\frac{65t}{65}$ | Divide, to isolate $t$ . | $\frac{d}{r}=\frac{rt}{r}$ | ||
Simplify. | $\phantom{\rule{1.2em}{0ex}}8=t$ | Simplify. | $\frac{d}{r}=t$ |
We say the formula $t=\frac{d}{r}$ is solved for $t$ .
Solve the formula $d=rt$ for $r$ :
ⓐ when $d=180\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t=4$ ⓑ in general
ⓐ $r=45$ ⓑ $r=\frac{d}{t}$
Solve the formula $d=rt$ for $r$ :
ⓐ when $d=780\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t=12$ ⓑ in general
ⓐ $r=65$ ⓑ $r=\frac{d}{t}$
Solve the formula $A=\frac{1}{2}bh$ for $h$ :
ⓐ when $A=90$ and $b=15$ ⓑ in general
ⓐ when $A=90$ and $b=15$ | ⓑ in general | ||||
Write the formula. | Write the formula. | ||||
Substitute. | |||||
Clear the fractions. | Clear the fractions. | ||||
Simplify. | Simplify. | ||||
Solve for $h$ . | Solve for $h$ . |
We can now find the height of a triangle, if we know the area and the base, by using the formula $h=\frac{2A}{b}$ .
Use the formula $A=\frac{1}{2}bh$ to solve for $h$ :
ⓐ when $A=170$ and $b=17$ ⓑ in general
ⓐ $h=20$ ⓑ $h=\frac{2A}{b}$
Use the formula $A=\frac{1}{2}bh$ to solve for $b$ :
ⓐ when $A=62$ and $h=31$ ⓑ in general
ⓐ $b=4$ ⓑ $b=\frac{2A}{h}$
The formula $I=Prt$ is used to calculate simple interest, I , for a principal, P , invested at rate, r , for t years.
Solve the formula $I=Prt$ to find the principal, $P$ :
ⓐ when $I=\text{\$}\mathrm{5,600},r=4\%,t=7\phantom{\rule{0.2em}{0ex}}\text{years}\phantom{\rule{0.2em}{0ex}}$ ⓑ in general
ⓐ $I=\mathrm{\$5,600}$ , $r=\mathrm{4\%}$ , $t=\mathrm{7\; years}$ | ⓑ in general | ||||
Write the formula. | Write the formula. | ||||
Substitute. | |||||
Simplify. | Simplify. | ||||
Divide, to isolate P . | Divide, to isolate P . | ||||
Simplify. | Simplify. | ||||
The principal is |
Use the formula $I=Prt$ to find the principal, $P$ :
ⓐ when $I=\text{\$}\mathrm{2,160},r=6\%,t=3\phantom{\rule{0.2em}{0ex}}\text{years}\phantom{\rule{0.2em}{0ex}}$ ⓑ in general
ⓐ $12,000 ⓑ $P=\frac{I}{rt}$
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