Final answer:
To find the revenue function from the marginal revenue function MR = 170 - 12x^1/3, integrate MR with respect to x. The integration yields R(x) = 170x - 9x^4/3, considering revenue is zero when no units are produced.
Step-by-step explanation:
The student has asked for assistance in finding the revenue function from the given marginal revenue function, which is MR = 170 - 12x1/3.
To find the revenue function R(x), we need to integrate the marginal revenue function with respect to x. Since the revenue is zero when no units are produced, we will also solve for the constant of integration C. The indefinite integral of MR with respect to x is:
\( R(x) = \int MR dx = \int (170 - 12x^{1/3}) dx \)
By integrating, we get:
\( R(x) = 170x - 12 \cdot \frac{3}{4}x^{4/3} + C \)
\( R(x) = 170x - 9x^{4/3} + C \)
To find C, we set R(0) to be 0, since the revenue is zero when nothing is produced. Solving for C gives us C = 0. Therefore, the revenue function is:
\( R(x) = 170x - 9x^{4/3} \)