Answer:
The marginal cost function is given by:
[tex]C'(x) = \frac{x^2}{5} +8[/tex] ......[1]
Now, we can find the cost function by taking the antiderivative :
[tex]C(x) = \int C'(x)[/tex] ......[2]
Substitute [1] in [2] we get;
[tex]C(x) = \int \frac{x^2}{5} +8 [/tex]
Integration formulas:
then;
[tex]C(x) = \frac{x^3}{5 \cdot 3} +8x + c_1[/tex] where [tex]c_1[/tex] is constant
Simplify:
[tex]C(x) = \frac{x^3}{15} +8x + c_1[/tex] ......[3]
It is also given that for 32 units costs is $ 381
substitute the value of x = 32 and C(x) = $ 381, in [3] to find the constant term [tex]c_1[/tex]
[tex]C(32) = \frac{(32)^3}{15} +8(32)+ c_1[/tex] or
[tex]381 = \frac{(32)^3}{15} +8(32)+ c_1[/tex]
or
[tex]381 = 2184.53333 + 256 +c_1[/tex]
[tex]381 = 2440.53333+c_1[/tex]
Simplify:
[tex]c_1 \approx -2059.53[/tex]
Substitute this value in [3] we get;
[tex]C(x) = \frac{x^3}{15} +8x - 2059.53[/tex]
Therefore, the cost function is, [tex]C(x) = \frac{x^3}{15} +8x - 2059.53[/tex]