A study of administrative intensity (the number of managers divided by the number of workers) found that the profit P of a company depended on the administrative intensity i according to the formula P(i) = (p - r)AS^ai^tk^bX^1-c - W₀ X - W₁Xi - C, where p is the selling price of a product; r is the nonwage cost of the product; A is a productivity scale factor; S is the number of occupational specialties; a, b, and t are positive numbers whose sum is less than 1 representing the marginal relative productivity of specialization, capital, and labor, respectively; k is the capital intensity per worker; X is the number of workers; c = 1 - a - b - t; W0 is the average wage per worker; W₁ is the average salary per manager; and C is the fixed cost of production.

Suppose that

(a) Show that the profit function simplifies to P(i) = 150√i - 300i - 10.
(b) Find the administrative intensity that maximizes profit.

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(pr)AS"kbX¹-c = 150, W₁X + C = 10, W₁X 300, and t = 0.5. =