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

Question

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

Answer 1

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

Answer 2

To maximize Mark's profits, he needs to find the optimal allocation of funds for production and advertising expenses, given a total budget of P1.5 million.

Let's assume Mark spends x amount on production and y amount on advertising. Since his profit varies as the product of these two amounts, we can represent his profit equation as:

Profit = xy^(1/2)

Now, we need to set up the problem to find the maximum profit. However, before proceeding, it's essential to clarify if there are any other constraints or limitations that need to be considered. If there aren't any other constraints mentioned, we can proceed with the maximization problem.

To find the maximum, we can use calculus. By taking the partial derivatives of the profit function with respect to both x and y, we can identify critical points. Setting these partial derivatives equal to zero will provide the points where the profit is maximized.

First, let's calculate the partial derivative of the profit function with respect to x:

d(Profit)/dx = y^(1/2)

Next, let's calculate the partial derivative of the profit function with respect to y:

d(Profit)/dy = (1/2)xy^(-1/2)

Setting these partial derivatives equal to zero, we get:

y^(1/2) = 0 (Equation 1)
(1/2)xy^(-1/2) = 0 (Equation 2)

However, Equation 2 does not have any practical solutions since x cannot be zero (assuming that both production and advertising expenses need to be positive).

Therefore, we need to focus on Equation 1, y^(1/2) = 0.

To find the optimal allocation, we need to solve this equation. However, y^(1/2) = 0 only when y = 0. This means that Mark should not spend any money on advertising to maximize his profit.

Given that the total budget is P1.5 million, Mark should allocate the entire budget towards the production expenses.

In summary, to maximize profits, Mark should allocate the entire budget of P1.5 million to production and not spend any money on advertising.