A linear revenue function is \( R=41 x \). (Assume \( R \) is measured in dollars.) (a) What is the slope \( m \) ? (b) What is the marginal revenue \( \overline{M R} \) ? What does the marginal revenue mean? Each additional unit sold yields this many dollars in revenue. Each additional unit sold decreases the revenue by this many dollars. If the number of units sold is increased by this amount, the revenue de (c) What is the revenue received from selling one more item if 50 are current \( \$ \square \) units sold is increased by this amount, the revenue inc What is the revenue received from selling one more item if 100 are being sold \( \$ \square \)

To analyze the linear revenue function \( R = 41x \), where \( R \) is the revenue in dollars and \( x \) is the number of units sold, we can answer the questions as follows: ### (a) What is the slope \( m \)? The slope \( m \) of the linear function \( R = 41x \) is the coefficient of \( x \). Therefore, the slope is: \[ m = 41 \] ### (b) What is the marginal revenue \( \overline{M R} \)? The marginal revenue \( \overline{M R} \) is the rate of change of revenue with respect to the number of units sold. In a linear revenue function, the marginal revenue is equal to the slope of the revenue function. Thus: \[ \overline{M R} = 41 \] **What does the marginal revenue mean?** Marginal revenue represents the additional revenue generated from selling one more unit of the product. In this case, it means: - Each additional unit sold yields **$41** in revenue. ### (c) What is the revenue received from selling one more item if 50 units are currently sold? To find the revenue received from selling one more item when 50 units are sold, we can calculate the revenue for 50 units and then for 51 units: 1. Revenue for 50 units: \[ R(50) = 41 \times 50 = 2050 \] 2. Revenue for 51 units: \[ R(51) = 41 \times 51 = 2091 \] 3. Revenue from selling one more item: \[ R(51) - R(50) = 2091 - 2050 = 41 \] Thus, the revenue received from selling one more item if 50 units are currently sold is: \[ \$41 \] ### What is the revenue received from selling one more item if 100 are being sold? Similarly, we can calculate the revenue for 100 units and then for 101 units: 1. Revenue for 100 units: \[ R(100) = 41 \times 100 = 4100 \] 2. Revenue for 101 units: \[ R(101) = 41 \times 101 = 4141 \] 3. Revenue from selling one more item: \[ R(101) - R(100) = 4141 - 4100 = 41 \] Thus, the revenue received from selling one more item if 100 units are currently sold is: \[ \$41 \] ### Summary of Answers: - (a) Slope \( m = 41 \) - (b) Marginal revenue \( \overline{M R} = 41 \) - (c) Revenue from selling one more item if 50 units are sold: \( \$41 \) - Revenue from selling one more item if 100 units are sold: \( \$41 \)