You purchase 600 glow sticks and plan to sell them at a Star Wars convention.
Your profit, P, in dollars from selling x glow sticks is given by the equation P=1.5x-81.
Use this information to complete the table below and answer the questions that follow.
Looking at the table, notice how the first row give the x-value or the number of glow
sticks sold.
And the second row gives the P-value, which is the profit in dollars, which means each
column in the table represents an ordered pair that will satisfy the given equation.
Focusing on this first column here, we're given x = 0.
We need to determine the corresponding P value or the corresponding profit.
To do this we substitute 0 for x in the profit equation.
So when x = 0, the profit, P, is equal to 1.5 times 0 minus 81.
1.5 times 0 is 0.
This simplifies to 0 minus 81, which equals negative 81.
This gives us the ordered pair 0 comma negative 81.
This means when 0 glow sticks are sold, the profit is negative 81 dollars, meaning there
is a loss of $81.
In the next column, we're given x = 28.
We need to find the corresponding P-value.
And therefore we substitute 28 for x.
When x = 28, P equals 1.5 times 28 minus 81.
Simplifying 1.5 times 28 equals 42.
This simplifies to 42 minus 81 which equals negative 39.
The ordered pair is 28 comma negative 39, which means if 28 glow sticks are sold, there's
a profit of negative $39, which means there's a loss of $39.
The next 2 columns in the table are different because we're given the value of P and asked
to determine the x-value.
So for this next column here, we're given the profit is 0.
We need to determine the value of x, which is the number of glow sticks sold.
So if P = 0, we substitute 0 for P in the profit equation, which gives us 0 equals 1.5
x - 81.
And now we need to solve this equation for x.
The first step is to isolate the x term, to do this we undo the subtraction by adding
81 to both sides of the equation.
Simplify, on the left we have 81 and equals on the right 1.5x.
-81 + 81 is 0.
1.5x means 1.5 times x and therefore to solve for x we divide both sides by 1.5 . 81 divided
by 1.5 equals 54.
On the right, 1.5 divided by 1.5 is 1.
1 times x is x.
When P = 0, x = 54, which means if 54 glow sticks are sold, the profit is $0.
In the next column, we're given P = 219.
We need to determine the value of x.
So if P = 219, we substitute 219 for P in the profit equation, giving us the equation
219 equals 1.5x minus 81.
To solve for x, we follow the same steps we used to solve for x in this equation.
First step, add 81 to both sides.
Simplify.
This gives us 300 equals 1.5x.
This simplifies to 0.
To solve for x we divide both sides by 1.5.
Simplifying, 300 divided by 1.5 is 200.
200 equals 1.5 divided by 1.5 is 1 and 1 times x is x. x = 200.
This tells us that if 200 glow sticks are sold, the profit is $219.
In the next column we are given x = 296.
We need to determine the corresponding value of P or corresponding profit.
When x = 296, the profit P equals 1.5 times 296 minus 81.
Simplifying, this product is equal to 444.
This simplifies to 444 minus 81, which equals 363.
When x = 296, P = 363, which means if 296 glow sticks are sold, the profit is $363.
For the last column in the table, we are given x = 600.
We to determine the corresponding value of P. When x = 600, the profit P equals 1.5 times
600 minus 81.
1.5 times 600 equals 900.
This simplifies to 900 minus 81, which equals 819.
When 600 glow sticks are sold, which would be all the glow sticks purchased, the profit
is $819.
And now let's answer the questions.
Interpret the meaning of the ordered pair 442 comma 582.
Well the first value in the ordered pair is x.
The second value in the ordered pair is P.
The meaning of this ordered pair is if you sell 442 glow sticks, the profit will be $582.
Next how many glow sticks must be sold to break even.
Break even means there is no loss or no profit or the profit is equal to 0.
So going back to the table, we already found the break even point.
Notice how this column here represents the break even point because P the profit is equal
to 0.
So the break even point is the ordered pair 54 comma 0, which means to break even you
must sell 54 glow sticks.
I hope you found this helpful.