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?
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.