Compute the maximum product for two positive numbers with the property that the sum of the first plus five times the second is 5000.
A) 25,000
B) 1,250,000
C) 600
D) 50,000
E) none of these

Question

Quizplus · Accepted Answer

Let the two positive numbers be $x$ and $y$, where $y$ is five times larger than $x$. We have the equation $x+5y=5000$. Solving for $x$ in terms of $y$ gives us $x=5000-5y$. We want to maximize the product $xy$.
$$xy=(5000-5y)y=-5y^2+5000y$$
This is a quadratic function with a negative leading coefficient, so its vertex gives the maximum value. The $y$-coordinate of the vertex is $\frac{-b}{2a}=\frac{-5000}{2(-5)}=500$, which corresponds to $x=5000-5(500)=2500$. Therefore, the two numbers are $2500$ and $1250$, and their product is $\boxed{	extbf{(B) }1,250,000}$.

Compute the Maximum Product for Two Positive Numbers with the Property