Question

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

Answer 1

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=(200)/(75)`

`\Rightarrow x=2(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