Marginal revenue Suppose that the revenue from selling x washing machines is r(x) = 20000(1 - 1/x) dollars. a. Find the marginal revenue when 100 machines are produced. b. Use the... Marginal revenue Suppose that the revenue from selling x washing machines is r(x) = 20000(1 - 1/x) dollars. a. Find the marginal revenue when 100 machines are produced. b. Use the function r'(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week. c. Find the limit of r'(x) as x → ∞. How would you interpret this number?
Understand the Problem
The question is asking for three specific tasks related to the function that describes the revenue from selling washing machines. The tasks include finding the marginal revenue for a specific quantity of machines, estimating the increase in revenue when production is slightly increased, and analyzing the behavior of the marginal revenue as production becomes very large.
Answer
a. $2$; b. $2$; c. $0$.
Answer for screen readers
a. The marginal revenue when 100 machines are produced is $2$ dollars.
b. The estimated increase in revenue when increasing production from 100 to 101 machines is approximately $2$ dollars.
c. The limit of $r'(x)$ as $x \to \infty$ is $0$.
Steps to Solve
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Define the Revenue Function The revenue function is given as: $$ r(x) = 20000 \left( 1 - \frac{1}{x} \right) $$
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Find the Marginal Revenue To find the marginal revenue, we need to differentiate the revenue function with respect to $x$: $$ r'(x) = \frac{d}{dx}\left[20000 \left( 1 - \frac{1}{x} \right)\right] $$
Applying the quotient rule to differentiate $\frac{1}{x}$, we get: $$ r'(x) = 20000 \cdot \left(0 + \frac{1}{x^2}\right) = \frac{20000}{x^2} $$
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Calculate Marginal Revenue at $x=100$ Now, plug in $x = 100$ into the marginal revenue function: $$ r'(100) = \frac{20000}{100^2} = \frac{20000}{10000} = 2 $$
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Estimate the Increase in Revenue from 100 to 101 Machines The estimated increase in revenue from producing one additional machine is: $$ r'(100) \approx 2 \text{ dollars} $$
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Find the Limit of Marginal Revenue as $x \to \infty$ We will calculate the limit of $r'(x)$ as $x$ approaches infinity: $$ \lim_{x \to \infty} r'(x) = \lim_{x \to \infty} \frac{20000}{x^2} = 0 $$
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Interpret the Limit As the production of washing machines increases indefinitely, the marginal revenue approaches zero, indicating that additional units contribute less to revenue.
a. The marginal revenue when 100 machines are produced is $2$ dollars.
b. The estimated increase in revenue when increasing production from 100 to 101 machines is approximately $2$ dollars.
c. The limit of $r'(x)$ as $x \to \infty$ is $0$.
More Information
Marginal revenue represents the additional revenue generated from selling one more unit of a product. In this case, as more washing machines are produced, the additional revenue gained from each individual machine decreases, approaching zero at large production levels. This is common in economics where saturation can occur in the market.
Tips
- Incorrect differentiation: Failing to correctly apply the quotient rule can lead to errors in finding $r'(x)$.
- Misinterpretation of the limit: Confusing the limit approaching zero with no revenue can lead to misunderstanding the implications of marginal revenue.
