A manufacturer finds it costs him x^2 +5x +7 dollars to produce x tons of an item. At production levels above 3 tons, he must hire additional workers, and his costs increase by 3(x-3) dollars on his total production. If the price he receives is #13 per ton regardless of how much he manufacturers and if he has a plant capacity of 10 tons, what level of output maximizes profits?
2007-10-25 10:02:56 · 2 answers · asked by wongtongsoup22 2 in Science & Mathematics ➔ Mathematics
P = 13x - x^2-5x-7 If x =/< 3
P = - x^2+ 8x-7 If x =/< 3
dP/dx = -2x+ 8 --> x = 4 so max would be at 3 tons.
P = -9 + 24 -7 = 8 dollars when x = 3
P = 13x -x^2 -5x -7 -3x + 9 if x>3
P = -x^2 +5x +2 if x>3
dP/dx = -2x +5 --> x = 2.5 which is less than 3
So it appears max occurs when x = 3 tons
2007-10-25 12:21:43 · answer #1 · answered by ironduke8159 7 · 0⤊ 0⤋
The cost function is
C(x)=x^2 +5x+7 for 0<=x<=3 and
x^2+8x-2 for 3<=x<=10
So the profit function is
P(x)=13x-C(x) and we are
looking for its maximum
P(x) is continous in[0,10] and taking
the zero of its derivative and
checking also what happens
at x=0,x=3,x=10 we find that
max occurs for x=3 tons
2007-10-25 10:41:05 · answer #2 · answered by katsaounisvagelis 5 · 0⤊ 0⤋