The deal your assistant signs calls for the sale of a minimum of 250 chairs and up to 440 chairs. the price will be $85 per chair if only 250 chairs are bought, but will be discounted by $0.2 per chair (on the entire order) for every chair ordered in addition to the minimum.
answer the questions below, rounding your answers to the nearest whole dollar.
what is the largest and least revenue you can make under this deal?

The largest revenue you can make is $22781 and you will make it if you sell 338 chairs.

The least revenue you can make is $20680 and you will make it in case you sell 440 chairs.

Step-by-step explanation:

Hi there!

The revenue is calculated by multiplying the number of chairs sold times the price of each chair. Let y be the revenue:

y = 250 · 85

If you sell more than 250 chairs, the price per chair will be discounted by $0.2 for every additional chair sold. So, if you sell x chairs more than 250, the discount per chair will be 0.2x. Then, the revenue, in this case, will be:

y = (250 + x) (85 - 0.2x)

Apply distributive property:

y = 21250 - 50x + 85x - 0.2x²

y = -0.2x² + 35x + 21250

The function is a negative parabola so that the largest revenue will be made at the vertex of the parabola. It is calculated using this formula:

x = -b/2a

Where:

a = -0.2

b = 35

Then:

x = -35/(2 · -0.2)

x = 87.5

Let´s round x to 88 because you can´t sell a half-chair.

Then , the maximum revenue will be:

y = -0.2x² + 35x + 21250

y = -0.2(88)² + 35(88) + 21250 = 22781

The largest revenue you can make is $22781 and you will make it if you sell 250 + 88 = 338 chairs.

Since the revenue function is a negative parabola the least revenue will be made at the extremes of the function (x = 0 and x = 440 -250 = 190). See the attached figure to notice this.

At x = 0

y = 21250

At x = 190

y =  -0.2(190)² + 35(190) + 21250 = 20680

The least revenue you can make is $20680 and you will make it in case you sell 440 chairs.