Quadratic Word Problems

Question

Katie makes and sells t-shirts. Her profit can be modeled by the function P(x) = -25x^2 + 625x - 2500, where x is the selling price of the shirt and P (x) is the profit made for each different price of the shirt.

a. What price shirt will produce the maximum profit?

b. How much profit can she make if she sells her shirts for $6 each?

c. What are the break-even selling prices for the t-shirts?

Answer 1

Since this equation has an x2 term, it is a quadratic function and its graph is a parabola. 

Since the first term is negative, it opens down, looking like an upside down letter u, and its vertex is a maximum. 

This maximum is a coordinate where x will be the t-shirt price and y will be the associated profit. 

Beginning with P(x) = -25x2 + 625x - 2500 

I would factor out -25 to keep the numbers more manageable

P(x) = -25(x2 - 25x +100)

Now factor x2 - 25x +100 to find the two x-intercepts 

P(x) = -25(x-20) (x-5)  

Solving x-20 = 0 The first x-intercept is (20,0)

Solving x-5 = 0 The second x-intercept is (5,0)

The vertex is located halfway between the two x-intercepts, so you can average the two x-intercepts to find the x coordinate of the vertex:

 (20+5)/2 =12.5

And you can use the original formula to find the associated profit:

P(12.5) = -25(12.5)2 + 625(12.5) - 2500 = -3906.25 + 7812.5 - 2500 = $1406.25

The maximum profit will be when the t-shirt price is $12.50, with profit of $1406.25

To find the profit when the t-shirt price is $6, use this price as input into the original formula:

P(6) = -25(6)2 + 625(6) - 2500 = -900 + 3750 - 2500 = $350

The profit when the t-shirt price is $6 is $350.

 

The break-even t-shirt price will be when the sales just covers the cost and profit is $0

We can use the same information we used to find the vertex, and set the equation = 0: 

P(x) = -25(x-20) (x-5) = 0 

Solving x-5=0 , we get $5 for t shirts

P(5) = -25(5)2 + 625(5) - 2500 = -625 + 3125 - 2500 = $0

Solving x-20=0 , we get $20 for t shirts

P(20) = -25(20)2 + 625(20) - 2500 = -10000 + 12500 - 2500 = $0

There are two break-even t-shirt prices, $5 and $20

 

Answer 2

P(x) = -25x^2 +625x -2500

P'(x) = -50x +625 = 0

x = 625/50 = 62.5/5 = 12.5

P(12.5) = -25(12.5)^2+625(12.5)- 2500

= -3,906.25 + 7,812.5 -2,500

= $1,406.25 = max profit when price= $12.50

P(6) = -25(36)+625(6) -2500 = -900+3750-2500= 1250-900

=$350 = profit when Price = $6

breakeven is when P(x)=0 = -25x^2+625x -2500, when profit = zero

x^2 - 25x +100 = 0

(x-20)(x-5) = 0

x= $5 or $20

max profit when price = average of breakeven prices = (5+20)/2 = $12.50