# Math

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Mark found out that his profit varies as the product of the amount spent for production and the square root of the amount spent for advertising. If his total available budget for these expenses is P1.5 million, how should he allocate his funds to maximize his profits

• Math -

If p is spent on production, and a on advertising, and the total budget is n,

P = kp√a
p+a <= n, so p = n-a
P = k(n-a)^2 √a
dP/da = -2k(n-a)√a + k(n-a)^2/(2√a)
= k(n-a)(n-5a)/(2√a)

So, maximum profit occurs when 1/5 of the budget is allocated to advertising

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