Question

Suppose yvaries directly as x. Write a direct variation equation that relates x and y. Then solve.

Suppose you earn $127 for working 20 hours.

a. Write a direct variation equation relating your earning to the number of hours worked.

b. How much would you earn for working 35 hours?

Short answer

a. The direct variation equation relating the earning to the number of hours worked is $y=\frac{127}{20}x$, where y represents the earning and x represents the number of hours worked.

b. The amount earned after working for 35 hours is 222.25.

Step-by-step solution

Part a. Step 1. Write the general equation of direct variation.

Let y represents the earning and xrepresents the number of hours worked.

It is given that earning varies directly to the number of hours worked.

Therefore, it implies that $y\alpha x$.

Therefore, it is obtained that:

$\begin{array}{l}y\alpha x\\ y=kx\end{array}$

Where k is constant of proportionality.

Part a. Step 2. Write a direct variation equation that relates the earning to the number of hours worked.

It is given that the earning earned after working for 20 hours is $127.

That implies, it is given that when $x=20,y=127$.

Therefore, substitute 20 for x and 127 for y in the equation$y=kx$ to find the value of k.

$\begin{array}{c}y=kx\\ 127=k\left(20\right)\\ \frac{127}{20}=k\end{array}$

Substitute the value of k in the equation $y=kx$.

Therefore, it is obtained that:

$\begin{array}{c}y=kx\\ y=\frac{127}{20}x\end{array}$

Therefore, the direct variation equation relating the earning to the number of hours worked is $y=\frac{127}{20}x$, where y represents the earning and x represents the number of hours worked.

Part b. Step 1. Find the value of y when x=35.

The direct variation equation that relates x and y is $y=\frac{127}{20}x$.

Find the value of y by substituting 35 for x in the equation $y=\frac{127}{20}x$.

$\begin{array}{c}y=\frac{127}{20}x\\ =\frac{127}{20}\left(35\right)\\ =222.25\end{array}$

Therefore, the value of y when$x=35$ is 222.25.

Part b. Step 2. Find the amount earned after working for 35 hours.

As, y represents the earning and x represents the number of hours worked.

Therefore, in order to find the amount earned after working for 35 hours, the value of y is to be calculated when $x=35$.

The value of y when$x=35$ is 222.25.

Therefore, the amount earned after working for 35 hours is 222.25.