(a) Marginal revenue:

(b) Revenue from the next thousand items at

To find the marginal revenue and the revenue from the next thousand items given the demand function
, we'll proceed step by step.
(a) Find the Marginal Revenue:
The derivative of the revenue function with respect to the quantity sold is known as the marginal revenue (MR).
The revenue function (R) is found by multiplying the demand function by the quantity sold, (x):
![\[ R = x \cdot p \]](https://img.qammunity.org/2024/formulas/business/high-school/7gixomau7zt10ps5cc82o7iwyb2znu98my.png)
Given
, we can express the revenue function in terms of x:
![\[ R = x \cdot p = x \left(100 + (70)/(\ln x)\right) \]](https://img.qammunity.org/2024/formulas/business/high-school/jfrbvtt61oqjehhjviretkk3ucwwdqeb45.png)
Take the revenue function's derivative with respect to x to determine the marginal revenue:
![\[ (dR)/(dx) = (d)/(dx) \left(x \cdot p\right) = 100 - (70)/((\ln x)^2) \cdot (1)/(x) \]](https://img.qammunity.org/2024/formulas/business/high-school/22ureb7urg6jatul95duca5qfkq5pk6j2b.png)
This derivative represents the marginal revenue function.
(b) Find the Revenue from the Next Thousand Items at a Demand of 4000 (x = 4):
At x = 4, we want to find the revenue from the next thousand items. To find this, first, calculate the revenue for 4000 items (x = 4):
![\[ R = x \cdot p = 4 \cdot \left(100 + (70)/(\ln 4)\right) \]](https://img.qammunity.org/2024/formulas/business/high-school/nuppixqotwe7zxbsdnzb8u225sd7jexty7.png)
Then, find the revenue for 3000 items (x = 3) to determine the revenue from the next thousand items:
![\[ R_{\text{next thousand}} = R(4000) - R(3000) \]](https://img.qammunity.org/2024/formulas/business/high-school/slpl7gy0nmvcok4kxce6gujocu67br9c3k.png)
Solving these expressions will give us the revenue from the next thousand items at a demand of 4000.
If you'd like, I can help you calculate these derivatives and the specific revenue values for x = 4 and x = 3 to determine the revenue from the next thousand items.