Complete question is;
Keegan is printing and selling his original design on t-shirts. He has concluded that for x shirts, in thousands sold his total profits will be p(x) = -x³ + 4x² + x dollars, in thousands will be earned. How many t-shirts (rounded to the nearest whole number) should he print in order to make maximum profits? What will his profits rounded to the nearest whole dollar be if he prints that number of shirts?
Answer:
Number of t-shirts to make maximum profit = 2790 shirts
Maximum profit = $12,209
Step-by-step explanation:
From the question, we are given that the profit function is;
p(x) = -x³ + 4x² + x
For the maximum value of the profit function,
(dp/dx) = 0 and (d²p/dx²) < 0
Since, p(x) = -x³ + 4x² + x
Then,
(dp/dx) = -3x² + 8x + 1
at maximum point (dp/dx) = 0, thus;
-3x² + 8x + 1 = 0
Solving this using quadratic formula, the roots are;
x = -0.12 or 2.79
Also, (d²p/dx²) = -6x + 8
Now, let's put the roots of x into -6x + 8 and check for maximum value conditon;
at x = -0.12
(d²p/dx²) = -6(0.12) + 8 = 7.28 > 0
At x = 2.79
(d²p/dx²) = -6(2.79) + 8 = -8.74 < 0
Maximum has to be d²p/dx² < 0
So, the one that meets the condition is -8.74 < 0 at x = 2.79
Thus, the maximum of the profit function exists when the number of shirts, x = 2.79 (in thousands) = 2790
Now, the maximum profits that corresponds to this number of t-shirts of 2.79(in thousands) is obtained by putting 2.79 for x in the profit function;
So,
p(2.79) = -(2.79)³ + 4(2.79²) + 2.79
p(x) = -21.7176 + 31.1364 + 2.79
p(x) = 12.2088 (in thousand dollars) ≈ $12,209
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