# Quadratic Function Word Problem

#### TrueStar

1. The problem statement, all variables and given/known data

A company sells running shoes to dealers at a rate f $40 per pair if fewer than 50 pairs are ordered. If a dealer orders 50 or more pairs (up to 600), the price per pair is reduced at a rate of 4 cents times the number ordered. What size order will produce the maximum amount of money for the company? 2. Relevant equations Vertex Formula -- -(b/2a) 3. The attempt at a solution My experience from other problems like this is that the charge should be set as x and the amount of shoes sold should be y. In order to figure out the number of shoes sold (y), I can create two points...but I think this is where I'm confused. Initially I wanted to use (40,49) and (39.96, 51). However, the problem states the discount is a rate of 4 cents times the number ordered. For example if I ordered 100 shoes, I'd pay$36 for each one (100*.04 is 4).

If I get two good points, I could get a slope, set up my equation for y and multiply that by x. Then I can use the vertex formula. I'm just having trouble figuring out how to write up this equation.

Thank you!

#### Mark44

Mentor
It probably makes more sense to use variable names that suggest what they're being used for. x is fine for the number of shoes ordered, but instead of y, I would suggest using C, rather than y, and with the idea that C represents the cost per pair of shoes.

If a dealer orders x pairs of shoes, what will the cost per pair be? You'll need a function that has one definition for one set of x values, and another definition for the other set of x values.

The revenue (R seems like a natural choice) will be the number of pairs of shoes sold times the cost per pair.

#### HallsofIvy

1. The problem statement, all variables and given/known data

A company sells running shoes to dealers at a rate f $40 per pair if fewer than 50 pairs are ordered. If a dealer orders 50 or more pairs (up to 600), the price per pair is reduced at a rate of 4 cents times the number ordered. What size order will produce the maximum amount of money for the company? 2. Relevant equations Vertex Formula -- -(b/2a) 3. The attempt at a solution My experience from other problems like this is that the charge should be set as x and the amount of shoes sold should be y. In order to figure out the number of shoes sold (y), I can create two points...but I think this is where I'm confused. Initially I wanted to use (40,49) and (39.96, 51). However, the problem states the discount is a rate of 4 cents times the number ordered. For example if I ordered 100 shoes, I'd pay$36 for each one (100*.04 is 4).

If I get two good points, I could get a slope, set up my equation for y and multiply that by x. Then I can use the vertex formula. I'm just having trouble figuring out how to write up this equation.

Thank you!
You don't want to "figure out the number of shoes sold". You are told how to calculate the price, x, as a function of y the number of shoes sold: The "base" price is $40 per pair. If y> 50, "the price per pair is reduced at a rate of 4 cents times the number ordered" so the$40 price is reduced by 0.04y: the price for each pair of shoes is 40- 0.04y.
The amount of money brought in is the price of each pair of shoes multiplied by the number of shoes sold: (40- 0.04y)(y)= 40y- 0.04y2. You want to find the maximum value of that, by using that "vertex formula". (I used your choice for x and y- notice that x is a quadratic function of y.) Of course, the answer must be between y= 50 and y= 600. If the vertex is not between those values the maximum will be at one of those to values.

#### TrueStar

Thanks for explaining this for me. It makes sense now and it looks like 500 would produce the maximum amount of money. Thanks again!

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