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The distance a moving body travels, d , varies directly with time, t , it moves. A train travels 100 miles in 2 hours
ⓐ Write the equation that relates
d and
t .
ⓑ How many miles would it travel in 5 hours?
ⓐ ⓑ 250 miles
In the previous example, the variables c and m were named in the problem. Usually that is not the case. We will have to name the variables in the next example as part of the solution, just like we do in most applied problems.
The number of gallons of gas Eunice’s car uses varies directly with the number of miles she drives. Last week she drove 469.8 miles and used 14.5 gallons of gas.
| The number of gallons of gas varies directly with the number of miles driven. | |
| First we will name the variables. | Let
number of gallons of gas.
number of miles driven |
| Write the formula for direct variation. |
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| We will use in place of and in place of . |
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| Substitute the given values for the variables. |
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| Solve for the constant of variation. |
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| We will round to the nearest thousandth. |
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| Write the equation that relates and . |
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| Substitute in the constant of variation. |
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Notice that in this example, the units on the constant of variation are gallons/mile. In everyday life, we usually talk about miles/gallon.
The distance that Brad travels varies directly with the time spent traveling. Brad travelled 660 miles in 12 hours,
ⓐ ⓑ 220 miles
The weight of a liquid varies directly as its volume. A liquid that weighs 24 pounds has a volume of 4 gallons.
ⓐ ⓑ 78 pounds
In some situations, one variable varies directly with the square of the other variable. When that happens, the equation of direct variation is . We solve these applications just as we did the previous ones, by substituting the given values into the equation to solve for k .
The maximum load a beam will support varies directly with the square of the diagonal of the beam’s cross-section. A beam with diagonal 4” will support a maximum load of 75 pounds.
| The maximum load varies directly with the square of the diagonal of the cross-section. | |
| Name the variables. | Let
maximum load.
the diagonal of the cross-section |
| Write the formula for direct variation, where varies directly with the square of . |
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| We will use in place of and in place of . |
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| Substitute the given values for the variables. |
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| Solve for the constant of variation. |
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| Write the equation that relates and . |
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| Substitute in the constant of variation. |
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