Question

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Remember it costs $14 to make the cakes and you are selling them for $50 each. If you only spend the 200 hours making the cakes, what was your maximum total profit?

Answer 1

[AI-generated answer — Bot GPT 3.5]

To calculate the maximum profit, we need to determine the number of cakes that can be made in 200 hours.

Let's assume it takes x hours to make one cake. This means that in 200 hours, you can make 200/x cakes.

To maximize profit, we want to make the maximum number of cakes and sell them all.

Let's calculate the maximum number of cakes: 200/x = maximum number of cakes.

To find the optimal value for x, we need to determine the value of x that maximizes the number of cakes made.

Now, let's consider the cost and revenue involved in making and selling the cakes. It costs $14 to make one cake, and you will be selling each cake for $50.

The cost of making all the cakes would be: $14 * (200/x).
The revenue from selling all the cakes would be: $50 * (200/x).

The profit is calculated by subtracting the cost from the revenue: profit = revenue - cost.

Therefore, the maximum total profit is: (50 * (200/x)) - (14 * (200/x)).

Simplifying the expression, we get: (10000/x) - (2800/x).

To find the optimal value for x, we equate the derivative of the expression to zero.

Differentiating the expression, we get: -10000/x^2 + 2800/x^2 = 0.

Combining like terms, we get: -7200/x^2 = 0.

Rearranging the equation, we have: 7200/x^2 = 0.

Solving for x^2, we get: x^2 = 7200.

Taking the square root of both sides, we find: x = sqrt(7200) ≈ 84.85.

As we cannot have a fraction of a cake, we round the value of x to the nearest whole number, which is 85.

Now, we can substitute x = 85 into the expression for profit:

Profit = (10000/85) - (2800/85) = 117.65 - 32.94 = $84.71.

Therefore, the maximum total profit would be approximately $84.71, rounded to the nearest cent.