Questions: A manufacturer finds that the revenue generated by selling (x) units of a certain commodity is given by the function (R(x)=60 x-0.5 x^2), where the revenue (R(x)) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum? (), at units

A manufacturer finds that the revenue generated by selling (x) units of a certain commodity is given by the function (R(x)=60 x-0.5 x^2), where the revenue (R(x)) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

(), at units

Transcript text: 4. DETAILS MY NOTES $0 / 3$ Submissions Used A manufacturer finds that the revenue generated by selling $x$ units of a certain commodity is given by the function $R(x)=60 x-0.5 x^{2}$, where the revenue $R(x)$ is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum? $\$$ $\square$ , at $\square$ units Submit Answer

Solution

Solution Steps

Step 1: Identify the Coefficients

Given the revenue function $R(x) = ax - bx^2$, the coefficients are $a = 60$ and $b = 0.5$.

Step 2: Calculate the Number of Units for Maximum Revenue

The number of units to manufacture for maximum revenue is found using the formula $x = \frac{a}{2b}$. Substituting the given values, we get $x = \frac{60}{2 \times 0.5} = 60$.

Step 3: Calculate the Maximum Revenue

Substituting $x$ back into the revenue function, $R(x) = a\left(\frac{a}{2b}\right) - b\left(\frac{a}{2b}\right)^2$, we find the maximum revenue. Substituting the given values, we get $R(x) = 60\left(\frac{60}{2 \times 0.5}\right) - 0.5\left(\frac{60}{2 \times 0.5}\right)^2 = 1800$.