Question

A cruise ship is traveling at a rate of 20 miles per hour. How do you use a direct variation formula to calculate the number of miles traveled in 7 hours?

Answer 1

See explanation. The ship travels 140 miles in 7 hours.

Explanation:

A direct variation formula always looks something like this:

`y=kx`

where it is said that `y` varies directly with `x`, by a constant multiple `k`. When `x` goes up by 1, `y` goes up by `k`. In other words, whatever value `x` is, `y` is always `k` times as much as `x`:

`stackrel y y stackrel "is always"=stackrel k k stackrel "times as much as" times stackrel x x`

For this question, the number of miles traveled is in direct variation with the speed of the ship. That's because, if the number of hours it sails goes up by 1, the number of miles it travels goes up by 20 (hence 20 miles per hour). The direct variation formula for this relation is:

`[(y),("miles")]=[(20),("miles/hour")] times [(x),("hours")]`

or

`y=20x`

Using this formula, we can plug in any number of hours we like, and we'll get the corresponding distance the ship travels in that time. For instance, when `x=7`, we get:

`y=20x`
`color(white)y=20(7)`
`color(white)y=140`

So the ship travels 140 miles in 7 hours.