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?

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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?

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Answer 1 · 2017-11-12T13:59:28

3 hours

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

Explanation:

Given

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

Let the constant of variation be $d$ $d$

Stated that $t$ $t$ varies inversely with $r d \to d t = \frac{d}{r}$ $r color(white)("d") ->color(white)("d") t=d/r$

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

$t \times r d = d \frac{d}{r} \times r$ $color(green)(t color(red)(xxr)color(white)("d")=color(white)("d")d/rcolor(red)(xxr))$

$t r = d \times \frac{r}{r}$ $color(green)(tcolor(red)(r)=d xx color(red)(r)/r)$

But $\frac{r}{r}$ $r/r$ is the same as 1

$t r = d \times 1$ $tr=d xx 1$

$t r = d$ $tr=d$ turning this round the other way

$d = t r$ $d=tr$

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

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

So the initial distance driven $d$ $d$ is such that: $d = 2 \times 45 = 90$ $d=2 xx 45 = 90$

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

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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?

1 Answer

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