Mrs. Dellamorte starts a side-business selling her mixtape. She makes a weekly profit by selling 'x' mixtapes according to the equation p(x)=-0.5x²+40x-300

Question

How many mixtapes does she need to sell each week to maximize her profit? Don't forget to use units.

Answer 1

40 mixtapes per week

just take the derivative of the profit function and set it equal to 0

-x + 40 = 0

x = 40

that's assuming P(x) is a profit function. In microeconomics P(x) usually refers to price, not profit

IF P(x) = price, then you could maximize revenue by multiplying x by P(x) to get R(x) = xP(x)=

-0.5x^3 + 40x^2 -300x then take the derivative and set it equal to 0

R'(x) = -1.5x^2 +80x -300 = 0

where x= the profit maximizing revenue.

But maxing revenue is not the same as maximizing profit. Profit already includes cost, Profit=Revenue minus Cost. 80x does look like Price times quantity, with -0.5x^2 -300 the cost function, so 40 is "probably" the answer you want for x

Answer 2

Hi Mia V

40 mixtapes

Maximum weekly profit of $500.00

p(x)=-0.5x²+40x-300

Your function above is a parabola in standard from of

f(x) = ax² + bx + c

Where

a = -0.5

b = 40

c = 300

Since a is negative, your parabola will open downward, so the vertex would be the maximum and the coordinates would represent the x and y values needed. The x value gives the number of tapes and can be calculated as -b/2a

-40/2(-0.5) = 40 = the x coordinate of the vertex

p(40) = -0.5(40)² + 40(40) - 300

p(40) = 500

So you need to sell 40 mixtapes to obtain a maximum weekly profit of $500.00

You can check your class textbook or search online to confirm the formula for calculating the vertex from the Standard Form Equation for a parabola. You can graph your parabola at Desmos.com to confirm the vertex.

I hope this helps