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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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.
Cristian
4.7119 Answers
Let's denote the two numbers as x y x^2 \times y^3 x + y = 30 y = 30 - x
Now, we can express the product x^2 \times y^3
P = x^2 \times (30 - x)^3
To find the maximum product, we can differentiate P x
\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 x y y = 30 - x
Finally, we can calculate the maximum product:
\text{Product} = x^2 \times y^3
\textbf{Answer:} x = 10 y = 20 10^2 \times 20^3 = 8,000,000
Now, we can express the product
To find the maximum product, we can differentiate
Simplify:
Solving the above equation will give us the value of
Finally, we can calculate the maximum product:
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