# How to calculate minimum profit

• Jul 14th 2010, 11:02 AM
Angie80
How to calculate minimum profit
Provided my sales are reflected by the following function:
g(x)=-3/2x^2 +195x-3500

At what price should I sell my goods to make any profit at all??
• Jul 14th 2010, 11:12 AM
SpringFan25
you need to say what the costs are in order to calculate the profit.
• Jul 14th 2010, 11:21 AM
Angie80
Oh, OK. Here is the info

75 glasses of lemonade sold at 60 cents per day
45 glasses at 80 cents per day.

function of sale price p(x)= -3/2x +165

gross profit f(x)= -3/2x^2 +165
net profit g(x) = -3/2x^2 +195x-3500
• Jul 14th 2010, 11:24 AM
Angie80
Here is a problem I am working on...

Ann is staring up her lemonade stand. Number of glasses of lemonade sold per day is a function of sale price p=p(x), where x is the price.
She notices that 75 glasses of lemonade were sold at 60 cents, but only 45 at 80 cents. If number of glasses sold is a linear function of sale price, what should p(x) be?
So, I came up with the solution:
Linear function p(x) = -3/2x +165

and now:

The gross profit f(x) of her lemonade stand is equal to the number of glasses sold, multiplied by sale price. It costs 20 cents to get the ingredients for one glass of lemonade. It also costs 2 dollars a day to keep her brother from interfering with the business.
Sketch the gross profit f(x) and the overhead (as function of price) on the same graph. Find an expression for the net profit expressed as a function of the price.

I came up with something like this:
gross profit
f(x) = p(x)(x)
f(x) = (-3/2x +165 )x= -3/2x^2 + 165x

Net profit g(x)= -3/2x^2 +195x-3500

And now, at what price should lemonade be sold to make any profit at all??
• Jul 14th 2010, 12:12 PM
1005
Quote:

Originally Posted by Angie80
Here is a problem I am working on...

Ann is staring up her lemonade stand. Number of glasses of lemonade sold per day is a function of sale price p=p(x), where x is the price.
She notices that 75 glasses of lemonade were sold at 60 cents, but only 45 at 80 cents. If number of glasses sold is a linear function of sale price, what should p(x) be?
So, I came up with the solution:
Linear function p(x) = -3/2x +165

and now:

The gross profit f(x) of her lemonade stand is equal to the number of glasses sold, multiplied by sale price. It costs 20 cents to get the ingredients for one glass of lemonade. It also costs 2 dollars a day to keep her brother from interfering with the business.
Sketch the gross profit f(x) and the overhead (as function of price) on the same graph. Find an expression for the net profit expressed as a function of the price.

I came up with something like this:
gross profit
f(x) = p(x)(x)
f(x) = (-3/2x +165 )x= -3/2x^2 + 165x

Net profit g(x)= -3/2x^2 +195x-3500

And now, at what price should lemonade be sold to make any profit at all??

If you solve g(x) = 0 for x, you will calculate two prices for when the net profit is zero. Notice, the x^2 has a negative coefficient, meaning it opens down. If you cannot remember this rule, you can simply evaluate g(x) at three values: one before the first root, one in the middle of the two roots, and one after the last root. Now, since the parabola points down, a cost a little bit above the first root will be the lowest price that earns a net profit. Therefore, if the first root has fractions of a cent, round the first root to the closest, most positive cent, and that is your answer (you can not sell lemonade at fractions of a penny!). If the first root is a whole number of cents, add one cent to it (Otherwise, you would've answered "What is the lowest price for which she earns zero net profit").
• Jul 14th 2010, 12:30 PM
Angie80
Thank you very much! Very helpful!!!