Question

vinnys company customizes and sells hats the function p(x)=-10x^2+700x-6000 graphed below indicats how much profit he makes in a month as a function of selling price

what should vinny charge per hat to make the maximum profit and what is the maximum profit he can make

A 4850 at 25 per hat
B 6250 at 35 per hat
C 7000 at 30 per hat
D 6000 at 40 per hat

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the selling price that would maximize the profit, we need to find the x-coordinate of the vertex of the parabola given by the profit function.

The profit function is given by p(x) = -10x^2 + 700x - 6000.

To find the x-coordinate of the vertex, we can use the formula x = -b/2a, where a = -10 and b = 700.

x = -700/(2*-10) = -700/-20 = 35.

Therefore, Vinny should charge $35 per hat to maximize his profit.

To find the maximum profit, we substitute x = 35 into the profit function:

p(35) = -10(35)^2 + 700(35) - 6000 = -10(1225) + 24500 - 6000 = -12250 + 24500 - 6000 = 6250.

Therefore, the maximum profit Vinny can make is $6250.

So the correct answer is B) 6250 at 35 per hat.

Answer 2

[AI-generated answer — Explain Bot]

To find the selling price that would maximize Vinny's profit and the corresponding maximum profit, we can first analyze the given function p(x) = -10x^2 + 700x - 6000.

The selling price of each hat is represented by x, and p(x) represents the profit Vinny makes in a month based on the selling price. To find the selling price that maximizes the profit, we need to find the vertex of the quadratic function.

The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a, b, and c are the coefficients of the quadratic function in the form ax^2 + bx + c.

From the given function, we can see that a = -10 and b = 700. Plugging these values into the formula, we get:

x = -700 / (2 * -10)
x = 35

Therefore, the selling price that Vinny should charge per hat to maximize his profit would be $35.

To find the maximum profit, we can substitute the value of x back into the original function to get the corresponding profit:

p(35) = -10(35)^2 + 700(35) - 6000
p(35) = -10(1225) + 24500 - 6000
p(35) = -12250 + 24500 - 6000
p(35) = 6250

Therefore, the maximum profit Vinny can make is $6250.

Based on the available options, we can see that option B matches the calculated values: 6250 at 35 per hat.

Answer 3

[AI-generated answer — Step-by-Step Bot]

To find the maximum profit and the selling price per hat that Vinny should charge, we need to determine the vertex of the function p(x) = -10x^2 + 700x - 6000.

Step 1: Find the x-coordinate of the vertex.
To find the x-coordinate of the vertex, we use the formula x = -b / 2a, where the equation is in the form ax^2 + bx + c.

In this case, a = -10 and b = 700.
x = -700 / (2*(-10))
x = -700 / (-20)
x = 35

Step 2: Find the y-coordinate of the vertex.
To find the y-coordinate of the vertex, substitute the x-coordinate (35) into the equation.
p(35) = -10(35)^2 + 700(35) - 6000
p(35) = -10(1225) + 24500 - 6000
p(35) = -12250 + 24500 - 6000
p(35) = 2450

The vertex of the function is (35, 2450), which represents the maximum profit at a selling price of $35 per hat.

Now let's compare the options given:
A: At $25 per hat, the profit is given as $4850.
B: At $35 per hat, the profit is given as $6250.
C: At $30 per hat, the profit is given as $7000.
D: At $40 per hat, the profit is given as $6000.

The correct answer is B. Vinny should charge $35 per hat to make the maximum profit. The maximum profit he can make is $6250.