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
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.
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.
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.