The function P (t) = - 4t^2 + 32l - 52 gives the profit in thousands of producing l units of lip gloss. What is the maximum profit that can be made?

Answer 1

To find the maximum profit, we need to find the vertex of the parabola represented by the function P(t) = -4t^2 + 32t - 52.

The x-coordinate of the vertex can be found using the formula t = -b/(2a), where a = -4 and b = 32 in this case.

t = -32/(2 * -4)
t = -32/-8
t = 4

So, the maximum profit can be made when t = 4.

Now, we can find the maximum profit by plugging t = 4 back into the function P(t):
P(4) = -4(4)^2 + 32(4) - 52
P(4) = -4(16) + 128 - 52
P(4) = -64 + 128 - 52
P(4) = 64 - 52
P(4) = 12

Therefore, the maximum profit that can be made is $12,000.