You decide to sell snow cones a local park as a summer job. The snow cone
machine costs $120 per month to rent and it cost to make a snow cone is $0.25,
which means the cost equation is y equals 120 + at 0.25 x, where is the number of
snow cones sold and y is the cost in dollars. If you sell it snow cone for $1.50
determine the break even point each month. So if you sell each snow cone for $1.50,
this means that revenue equation is y equal 1.5 x, where again x is the number
of snow cone sold and y is the revenue in dollars. Now the break even point is
a point at which cost and revenue are equal, which means there is no loss or profit,
where the company doesn't lose money or make money. So to find the break even
point, which will be an ordered pair x comma y, we need to solve this as a system of
equations. Let's use the method of substitution to solve the system. Looking
at the second equation because we have y equals 1.5 x, let substitute 1.5 x or
Y in the first equation, which will give us one equation with just x.
Performing the substitution, we would have 1.5 x equals 120 plus 0.25 x.
And now to solve this equation, let's get the x terms on one side of the equation.
To undo plus 0.25 x, we subtract 0.25 x on both sides. Now we simplify. 1.5 x minus 0.25 x
is 1.25 x. So we have 1.25 x equals, on the right, plus 0.25 x minus 0.25 x is zero.
So the right side simplifies to 120. Now to solve for x we divide both sides by 1.25
Simplifying, this quotient is 1. 1 times x is x.
So we have x equals 120 divided by 1.25 equals 96. So now we know for the break even
point the x value is 96. We still need to find the y value. Because we know the x value is 96, this
means you must sell 96 snow cones each month in order to break even. And when x
equals 96, both the cost and revenue should be the same or should be equal . Let's
go ahead and verify this. Let's first check the cost when x equals 96. The cost would
be y equals 120 plus 0.25 times 96, which would be 120 plus, 0.25 times 96 is 24. And therefore we have a
cost of $144 when x equals 96. Now we should get the same amount for the
revenue when x equals 96. Again let's go and check this. Revenue is y
equals 1.5 times 96, which is equal to 144. Now we know the
break-even point is 96 comma 144. Let's write this is a sentence.
The break even point is 96 comma 144, which means if you sell 96 snow cones
per month, you won't lose money or make money,
which means
more than 96 snow cones you make a profit. If you sell less than 96 snow
cones you have a loss. Before we go let's look at this graphically. In blue
we have the cost equation. In red we have a graph of the revenue equation.
Notice how the point of intersection is your break even point 96 comma 144
So when you sell more than 96 snow cones, the revenue is more
than the cost. And therefore this region here represents the region when you have
a profit. When you sell less than 96 snow cones the cost is more than the
revenue and therefore this region here represents when you have a loss.
I hope you found this helpful.