Question

Algebra Word Problem, 9th Grade, Station 2, please help!

Answer 1

ANSWER

ANSWER TO QUESTION 1.

If you give $x$ surfing lessons, then you will earn $ $25x$

If you are charged $200 for using the place, then your profit will reduce by $200.

The profit function is given by

$P(x)=25x-200$ dollars.

The y-intercept is  $-200$. It is negative because it is a cost which is a liability. So it does not add to your profit.

ANSWER TO QUESTION 2.

The slope is $25$. It is positive because as the number of hours increases, the profit earned also increases. In other words there is a direct relation between the number of hours worked and the profit earned.

ANSWER TO QUESTION 3.

The slope intercept form is when an equation is written in the form;

$y=mx+c$,

where $m$ is the slope and $c$ is the y-intercept. The profit function in slope intercept form is

$P(x)=25x-200$

ANSWER TO QUESTION 4.

If you break even, then the difference between the revenue and cost is zero. So we equate the profit function to zero.

$25x-200=0$

$25x=200$

$x=8$

Therefore you need to give 8 different lessons to break even.

ANSWER TO QUESTION 5.

If you give 20 lessons, the $x=20$. We need to substitute

$x=20$ in to the profit function to calculate the profit made after 20 lessons.

$P(20)=25(20)-200$

$P(20)=500-200$

$P(20)=300$.

ANSWER TO QUESTION 6.

If the new price is $75, then the new profit function becomes

$P(x)=75x-200$

ANSWER TO QUESTION 7.

To find the number of lessons that lets you break even, we equate the new profit function to zero.

$75x-200=0$

We solve for x by adding 200 to both sides of the equation.

$\Rightarrow 75x=200$

We now divide through by 75

$\Rightarrow x=\frac{200}{75}$

$\Rightarrow x=2\frac{2}{3}$

Therefore you would have to 2\frac{2}{3} lessons to break even. That is approximately 3 lessons.

ANSWER TO QUESTION 8.

If you give 20 lessons and we want to find the profit you made with this new profit function, then we have to plug in $x=20$ in to $P(x)=75x-200$.

That is,

$P(20)=75(20)-200$

$P(20)=1500-200$

$P(20)=1300$ dollars