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Math  /  Calculus

QuestionIf the total revenue received from the sale of xx items is given by R(x)=10ln⁡(5x+1)R(x)=10 \ln (5 x+1), while the total cost to produce xx items is C(x)=5^5C(x)=\frac{\hat{5}}{5}, find the following. (a) The marginal revenue (b) The profit function P(x)\mathrm{P}(\mathrm{x}) (c) The marginal profit when x=50x=50 (d) Interpret the results of part (c). (a) How can the marginal revenue be tound? A. Find the derivative of R(x)−C(x)R(x)-C(x). B. Find the derivative of R(x)R(x). C. Find R(x)−C(x)R(x)-C(x). D. Find R(x2)R\left(\frac{x}{2}\right).
The marginal revenue is □\square (b) How can the profit function be found? A. Find the derivative of R(x)−C(x)R(x)-C(x). B. Find the derivative of R(x)R(x). C. Find R(x)−C(x)R(x)-C(x). D. Find R′(x)−C(x)R^{\prime}(x)-C(x).
The profit function is P(x)=\mathrm{P}(\mathrm{x})= □\square . (c) The marginal profit when x=50x=50 can be found by evaluating □\square The marginal profit when x=50x=50 is □\square (Type an integer or decimal rounded to the nearest tenth as needed.) (d) Interpret the results of part (c). A. Profit is negligible when producing fewer than 50 items. B. There is little cost to producing more than 50 items.

Studdy Solution

STEP 1

1. The revenue function is R(x)=10ln⁡(5x+1) R(x) = 10 \ln(5x + 1) .
2. The cost function is C(x)=55=1 C(x) = \frac{5}{5} = 1 .
3. We need to find the marginal revenue, profit function, and marginal profit at x=50 x = 50 .

STEP 2

1. Find the marginal revenue.
2. Determine the profit function.
3. Calculate the marginal profit when x=50 x = 50 .
4. Interpret the results of part (c).

STEP 3

To find the marginal revenue, we need to find the derivative of the revenue function R(x) R(x) .
The correct choice for finding the marginal revenue is: B. Find the derivative of R(x) R(x) .
Calculate the derivative: R′(x)=ddx[10ln⁡(5x+1)]=10⋅55x+1=505x+1 R'(x) = \frac{d}{dx}[10 \ln(5x + 1)] = \frac{10 \cdot 5}{5x + 1} = \frac{50}{5x + 1}
The marginal revenue is 505x+1\frac{50}{5x + 1}.

STEP 4

To find the profit function, we subtract the cost function from the revenue function.
The correct choice for finding the profit function is: C. Find R(x)−C(x) R(x) - C(x) .
Calculate the profit function: P(x)=R(x)−C(x)=10ln⁡(5x+1)−1 P(x) = R(x) - C(x) = 10 \ln(5x + 1) - 1
The profit function is P(x)=10ln⁡(5x+1)−1 P(x) = 10 \ln(5x + 1) - 1 .

STEP 5

To find the marginal profit when x=50 x = 50 , we need to evaluate the derivative of the profit function at x=50 x = 50 .
The marginal profit can be found by evaluating: P′(x)=R′(x)−C′(x) P'(x) = R'(x) - C'(x)
Since C(x)=1 C(x) = 1 , C′(x)=0 C'(x) = 0 .
Thus, P′(x)=R′(x) P'(x) = R'(x) .
Evaluate P′(50)=R′(50)=505(50)+1=50251≈0.1992 P'(50) = R'(50) = \frac{50}{5(50) + 1} = \frac{50}{251} \approx 0.1992
The marginal profit when x=50 x = 50 is approximately 0.2 0.2 .

STEP 6

Interpret the results of part (c).
The correct interpretation is: A. Profit is negligible when producing fewer than 50 items.
The marginal revenue is 505x+1\frac{50}{5x + 1}.
The profit function is P(x)=10ln⁡(5x+1)−1 P(x) = 10 \ln(5x + 1) - 1 .
The marginal profit when x=50 x = 50 is approximately 0.2 0.2 .
Interpretation: Profit is negligible when producing fewer than 50 items.

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