Question

Jerry and his family will drive 1,022 miles from Houston to Indianapolis for their summer vacation. They plan to drive between 425 and 475 miles each day. At this rate, which is a reasonable number of days it will take Jerry's family to reach Indianapolis?

Answer 1

The reasonable number of days to reach Indianapolis is 3 days.

Explanation:

Total distance to be covered from Houston to Indianapolis, d= 1022 miles.

As they plan to drive between 425 and 475 miles/day, so the range of the rate of driving per day, r, is 425<r<475.

Let t day is the reasonable number of days to complete the journey.

As time = (distance)/(speed), so

`t=\frac d r\cdots(i)`

The given range of the rate is

`425<r<475`

If
`a>b`, than
`1/a<1/b` for
`a,b\\eq0`;

`\Rightarrow \frac {1}{425}>\frac {1}{r} >(1)/(475)`

If a>b, the for any
`\alpha>0`,
`a\alpha>b\alpha`, so multiply the above equation by
`d`, we have

`\frac {d}{425}>\frac {d}{r} >(d)/(475)`

`\Rightarrow \frac {d}{425}>t >(d)/(475)` [from equation (i)]

`\Rightarrow \frac {1022}{425}>t >(1022)/(475)` [given, d=1022 miles]

`\Rightarrow \frac {1022}{425}>t >(1022)/(475)`

`\Rightarrow 2.40>t >2.15`

So, to cover the distance of 1022 miles, with the speed in between 425 miles/day and 475 miles/day, Jerry's family will take in between 2.40 days and 2.15 days. This means, for the given speed range, they need 2 complete days and some fraction of 3rd day to reach Indianapolis.

Hence, the reasonable number of days to reach Indianapolis is 3 days.