Answer:
t-shirts: 2790
profit: $12209
Step-by-step explanation:
Given the function:
p(x) = -x³ + 4x² + x
we want to maximize it.
The following criteria must be satisfied at the maximum:
dp/dx = 0
d²p/dx² < 0
dp/dx = -3x² + 8x + 1 = 0
Using quadratic formula:







d²p/dx² = -6x + 8
d²p/dx² at x = -0.12: -6(-0.12) + 8 = 8.72 > 0
d²p/dx² at x = 2.79: -6(2.79) + 8 = -8.74 < 0
Then, he should prints 2.79 thousands, that is, 2790 t-shirts to make maximum profits.
Replacing into profit equation:
p(x) = -(2.79)³ + 4(2.79)² + 2.79 = 12.209
that is, $12209
5.25 / .85 you would get 6.17~ but you cant sell 1/4th a cup, so you have to round up, making it 7 cups.
X=hours of plan A
y=hours of plan B
wed: 5x+6y=12
thu: 3x+2y=6
so
5x+6y=12 and
3x+2y=6
we can eliminate y's by multiplying 2nd equation by -3 and adding to first equation
-9x-6y=-18
<u>5x+6y=12 +</u>
-4x+0y=-6
-4x=-6
divid both sides by -4
x=-6/-4
x=3/2
x=1.5
sub back
3x+2y=6
3(1.5)+2y=6
4.5+2y=6
2y=1.5
y=0.75
Plan A=1.5 hours
Plan B=0.75 hours