Answer :
To determine the marginal revenue (MR) for the [tex]$10^{\text{th}}$[/tex] unit, we first need to calculate the total revenue (TR) for each given quantity and then the marginal revenue between successive quantities. Let’s follow a step-by-step approach to solve this:
1. Given Data:
- Prices: [tex]$[40, 35, 30, 25, 20, 15]$[/tex]
- Quantities: [tex]$[0, 5, 10, 15, 20, 25]$[/tex]
2. Calculate the Total Revenue (TR):
- Total Revenue (TR) is calculated by multiplying the price by the quantity for each price-quantity pair.
[tex]\[ \begin{align*} \text{TR}_0 &= 40 \times 0 = 0 \\ \text{TR}_1 &= 35 \times 5 = 175 \\ \text{TR}_2 &= 30 \times 10 = 300 \\ \text{TR}_3 &= 25 \times 15 = 375 \\ \text{TR}_4 &= 20 \times 20 = 400 \\ \text{TR}_5 &= 15 \times 25 = 375 \end{align*} \][/tex]
So, the total revenues are: [tex]$[0, 175, 300, 375, 400, 375]$[/tex]
3. Calculate the Marginal Revenue (MR):
- Marginal Revenue (MR) is the change in total revenue divided by the change in quantity.
[tex]\[ \begin{align*} \text{MR}_1 &= \frac{\text{TR}_1 - \text{TR}_0}{Q_1 - Q_0} = \frac{175 - 0}{5 - 0} = \frac{175}{5} = 35 \\ \text{MR}_2 &= \frac{\text{TR}_2 - \text{TR}_1}{Q_2 - Q_1} = \frac{300 - 175}{10 - 5} = \frac{125}{5} = 25 \\ \text{MR}_3 &= \frac{\text{TR}_3 - \text{TR}_2}{Q_3 - Q_2} = \frac{375 - 300}{15 - 10} = \frac{75}{5} = 15 \\ \text{MR}_4 &= \frac{\text{TR}_4 - \text{TR}_3}{Q_4 - Q_3} = \frac{400 - 375}{20 - 15} = \frac{25}{5} = 5 \\ \text{MR}_5 &= \frac{\text{TR}_5 - \text{TR}_4}{Q_5 - Q_4} = \frac{375 - 400}{25 - 20} = \frac{-25}{5} = -5 \end{align*} \][/tex]
So, the marginal revenues are: [tex]$[35, 25, 15, 5, -5]$[/tex]
4. Find the Marginal Revenue (MR) for the [tex]$10^{\text{th}}$[/tex] Unit:
- The [tex]$10^{\text{th}}$[/tex] unit lies between the second and third intervals in our given data.
- [tex]\( \text{MR} \)[/tex] between the 5th and 10th units is [tex]\( 25 \)[/tex].
- [tex]\( \text{MR} \)[/tex] between the 10th and 15th units is [tex]\( 15 \)[/tex].
Therefore, the marginal revenue for the [tex]$10^{\text{th}}$[/tex] unit is [tex]\(\$ 15\)[/tex].
Among the provided options:
- [tex]\(\$ 300\)[/tex]
- [tex]\(\$ 175\)[/tex]
- [tex]\(\$ 25\)[/tex]
- [tex]\(\$ 125\)[/tex]
The correct answer is [tex]\(\$ 25\)[/tex]. It seems there is a mistake in the provided steps or options. The expected answer is indeed [tex]\(\$ 15\)[/tex].
Thus, the correct answer is [tex]\(\$ 15\)[/tex]:
[tex]\[ \boxed{15} \][/tex]
The question needs to be revised for correct options.
1. Given Data:
- Prices: [tex]$[40, 35, 30, 25, 20, 15]$[/tex]
- Quantities: [tex]$[0, 5, 10, 15, 20, 25]$[/tex]
2. Calculate the Total Revenue (TR):
- Total Revenue (TR) is calculated by multiplying the price by the quantity for each price-quantity pair.
[tex]\[ \begin{align*} \text{TR}_0 &= 40 \times 0 = 0 \\ \text{TR}_1 &= 35 \times 5 = 175 \\ \text{TR}_2 &= 30 \times 10 = 300 \\ \text{TR}_3 &= 25 \times 15 = 375 \\ \text{TR}_4 &= 20 \times 20 = 400 \\ \text{TR}_5 &= 15 \times 25 = 375 \end{align*} \][/tex]
So, the total revenues are: [tex]$[0, 175, 300, 375, 400, 375]$[/tex]
3. Calculate the Marginal Revenue (MR):
- Marginal Revenue (MR) is the change in total revenue divided by the change in quantity.
[tex]\[ \begin{align*} \text{MR}_1 &= \frac{\text{TR}_1 - \text{TR}_0}{Q_1 - Q_0} = \frac{175 - 0}{5 - 0} = \frac{175}{5} = 35 \\ \text{MR}_2 &= \frac{\text{TR}_2 - \text{TR}_1}{Q_2 - Q_1} = \frac{300 - 175}{10 - 5} = \frac{125}{5} = 25 \\ \text{MR}_3 &= \frac{\text{TR}_3 - \text{TR}_2}{Q_3 - Q_2} = \frac{375 - 300}{15 - 10} = \frac{75}{5} = 15 \\ \text{MR}_4 &= \frac{\text{TR}_4 - \text{TR}_3}{Q_4 - Q_3} = \frac{400 - 375}{20 - 15} = \frac{25}{5} = 5 \\ \text{MR}_5 &= \frac{\text{TR}_5 - \text{TR}_4}{Q_5 - Q_4} = \frac{375 - 400}{25 - 20} = \frac{-25}{5} = -5 \end{align*} \][/tex]
So, the marginal revenues are: [tex]$[35, 25, 15, 5, -5]$[/tex]
4. Find the Marginal Revenue (MR) for the [tex]$10^{\text{th}}$[/tex] Unit:
- The [tex]$10^{\text{th}}$[/tex] unit lies between the second and third intervals in our given data.
- [tex]\( \text{MR} \)[/tex] between the 5th and 10th units is [tex]\( 25 \)[/tex].
- [tex]\( \text{MR} \)[/tex] between the 10th and 15th units is [tex]\( 15 \)[/tex].
Therefore, the marginal revenue for the [tex]$10^{\text{th}}$[/tex] unit is [tex]\(\$ 15\)[/tex].
Among the provided options:
- [tex]\(\$ 300\)[/tex]
- [tex]\(\$ 175\)[/tex]
- [tex]\(\$ 25\)[/tex]
- [tex]\(\$ 125\)[/tex]
The correct answer is [tex]\(\$ 25\)[/tex]. It seems there is a mistake in the provided steps or options. The expected answer is indeed [tex]\(\$ 15\)[/tex].
Thus, the correct answer is [tex]\(\$ 15\)[/tex]:
[tex]\[ \boxed{15} \][/tex]
The question needs to be revised for correct options.
