Q12. - Algebra 1

Question

The distance required for a car to stop is directly proportional to the square of its velocity. If a car stop in 242 meters at 22 kilometers per hour, how many meters are needed to stop at 30 kilometers per hour?

Step-by-Step Solution

Answer

To stop the car at the speed of 30 kilometers per hour 450 meters are needed.

Step 1. State the concept of Direct Proportionality.

The comparison of two numbers where their ratio gives a constant value is called as direct proportion. The direct proportionality symbol is ‘$\propto$’.

Suppose, if $y$ is directly proportional to $x$ then it is written as,

 $y\propto x$

To remove the proportionality symbol, a constant term say ‘$k$’ must be introduced.

That is, $y=kx$

Step 2. Find the value of constant ‘ k ’.

The car stops in 242 meters at 22 kilometers per hour.

Here, 242 meters is the distance and 22 kilometer per hour is the velocity of the car.

Let $y$ be the distance and $x$ be the velocity.

The distance required for the car to stop is directly proportional to the square of its velocity. 

That is,

$\begin{array}{l}y\propto x^2\\ y=k{x}^2 \dots \left(1\right)\end{array}$

Substitute $y=242$ and $x=22$ in $y=k{x}^2$ to find the value of $k$.

$\begin{array}{l}242=k{\left(22\right)}^2\\ 242=484k\\ \frac{242}{484}=\frac{484k}{484}\\ \frac{1}{2}=k\\ k=\frac{1}{2}\end{array}$

Step 3. Calculate the distance required to stop the car at 30 kilometer per hour speed.

Find the distance by substituting $x=30$ and $k=\frac{1}{2}$ in $y=k{x}^2$.

$\begin{array}{l}y=k{x}^2\\ =\left(\frac{1}{2}\right)\left({30}^2\right)\\ =\frac{1}{2}\times 900\\ =450\end{array}$

Therefore, 450 meters are needed to stop the car at 30 kilometer per hour.