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Find two non-negative numbers whose sum is 30 that make the maximum product of the square of one and the cube of the other.

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Answer to a math question Find two non-negative numbers whose sum is 30 that make the maximum product of the square of one and the cube of the other.

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Cristian

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119 Answers

Let's denote the two numbers as

and

. We need to find the maximum product of

x^2 \times y^3

subject to the constraint that

x + y = 30

. We can rewrite the constraint as

y = 30 - x

.

Now, we can express the product

x^2 \times y^3

in terms of one variable:

P = x^2 \times (30 - x)^3

To find the maximum product, we can differentiate

with respect to

and set the derivative equal to 0:

\frac{dP}{dx} = 2x \times (30 - x)^3 - 3x^2 \times 3 \times (30 - x)^2 = 0

Simplify:

2x \times (30 - x)^3 - 9x^2 \times (30 - x)^2 = 0

Solving the above equation will give us the value of

, and then we can find the corresponding value of

using

y = 30 - x

.

Finally, we can calculate the maximum product:

\text{Product} = x^2 \times y^3

\textbf{Answer:}

The two non-negative numbers that make the maximum product are

x = 10

and

y = 20

. The maximum product is

10^2 \times 20^3 = 8,000,000

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Find two non-negative numbers whose sum is 30 that make the maximum product of the square of one and the cube of the other.

Answer to a math question Find two non-negative numbers whose sum is 30 that make the maximum product of the square of one and the cube of the other.

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