Question

A highway patrolman spots a speeding car. He clocks it at 70 mph and takes after it 0.5 miles behind. If the patrolman travels at an average rate of 90 mph, how long before he overtakes the car?

Answer 1

Answer:

The patrolman overtakes the car after 1.5 minutes.

Step-by-step explanation:

The speed of the speeding car is 70 mph and the speed of the patrolman is 90 mph.

Suppose, the patrolman takes $t$ hour to overtake the car.

We know that,  $Distance= Speed*Time$

So, the distance traveled by the car in $t$ hour $= 70t$ miles

and the distance traveled by the patrolman in $t$ hour $= 90t$ miles.

Given that, the patrolman was 0.5 miles behind.

So, the equation will be......

$90t= 70t+0.5\\ \\ 90t-70t=0.5\\ \\ 20t=0.5\\ \\ t=\frac{0.5}{20}=0.025$

Now,  0.025 hours $=(0.025\times 60)minutes= 1.5$ minutes.

So, the patrolman overtakes the car after 1.5 minutes.

Answer 2

The patrolman overtakes the car after $\boxed{{\mathbf{1}}{\mathbf{.5 minutes}}}$ .

Further explanation:

It is given that a highway patrolman spots a speeding car.

The speed of the speeding car is given as $70{\text{ mph}}$  and the speed of the patrolman is $90{\text{ mph}}$ .

Consider the patrolman takes $t$  to overtake the speeding car.

Now, to calculate the distance covered by any object with speed $s$  and time $t$  is given below.

${\text{Distance}}=s\times t$

(1)

Substitute $70$ for $s$  in equation (1) to obtain the distance traveled by speeding car in $t$  hours.

${\text{Distance}}=70t$

Substitute $90$ for $s$  in equation (1) to obtain the distance traveled by patrolman in $t$  hours.

${\text{Distance}}=90t$

It is given that the patrolman was $0.5$  so the obtained equation is expressed as,

$90t=70t+0.5$  

Now, solve the above equation to obtain the value of $t$ .

$\begin{aligned}90t&=70t+0.5\\90t-70t&=0.5\\20t&=0.5\\t&=\frac{{0.5}}{{20}}\\\end{aligned}$

Further simplify the above equation.

$\begin{aligned}t&=\frac{5}{{200}}\\&=0.025{\text{ hrs}}\\\end{aligned}$

Therefore, the time is $0.025{\text{ hrs}}$ .

Now, in minutes it is converted as follows:

$\begin{aligned}t&=0.025\times60\\&=\frac{{25}}{{1000}}\times60\\&=\frac{{1500}}{{1000}}\\&=1.5\\\end{aligned}$

Therefore, the time is $1.5{\text{ min}}$ .

Thus, the patrolman overtakes the car after $\boxed{{\mathbf{1}}{\mathbf{.5 minutes}}}$ .

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Answer Details:

Grade: Junior High School

Subject: Mathematics

Chapter: Distance and Time

Keywords: Distance, time, speed, patrolman, overtakes, speeding car, car, $1.5{\text{ min}}$ , hours, overtakes the car after, $70{\text{ mph}}$ , speed.