Question

The time t required to drive a certain distance varies inversely with the speed r. If it takes 2 hours to drive the distance at 45 miles per hour, how long will it take to drive the same distance at 30 miles per hour?

Answer 1

3 hours

Solution given in detail so you can see where everything comes from.

Explanation:

Given

The count of time is `t`
The count of speed is `r`

Let the constant of variation be `d`

Stated that `t` varies inversely with `r color(white)("d") ->color(white)("d") t=d/r`

Multiply both sides by `color(red)(r)`

`color(green)(t color(red)(xxr)color(white)("d")=color(white)("d")d/rcolor(red)(xxr))`

`color(green)(tcolor(red)(r)=d xx color(red)(r)/r)`

But `r/r` is the same as 1

`tr=d xx 1`

`tr=d` turning this round the other way

`d=tr`

but the answer to `tr` ( time x speed ) is the same as distance
So `d` must be the distance.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Part 1 - Determine the distance traveled - initial condition")`

We are given that the initial time `t` is 2 yours
We are given that the initial speed `r` is 45 miles for each hour.

So the initial distance driven `d` is such that: `d=2 xx 45 = 90`

How do we handle the units of measurement. They behave the same way as do the numbers.

So we have:

`color(green)(d" miles"=color(red)(2cancel("hours")) xx color(purple)(45(" miles")/cancel("hours")) = 90" miles ")......Equation(1)`
Notice that the unit for hours cancels out leaving just miles

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Part 2 - Determine the new time for an increase in speed - new condition")`
Instead of writing miles use the letter `m`
Instead of writing hours use the letter `h`

So `Equation(1)` becomes:

`color(green)(dm=color(red)(2cancel(h)) xx color(purple)(45(m)/cancel(h)) = 90m)" "......Equation(1_a)`

In the new condition we do not know time so write `th`
The new speed is 30 miles per hour so write `30 m/h`
The distance traveled is the same so write `90m`

`color(green)(dm=color(red)(tcancel(h)) xx color(purple)(30(m)/cancel(h)) = 90m)" "......Equation(1_b)`

`txx30=90`

Multiply each side by `1/30`

`txx30/30=90/30`

`txx1=3`

`t=3`

But `t` is measured in hours so `t=3` hours