the price p (in dollars) and the demand x for a particular clock radio are related by the equation x = 5000…

the price p (in dollars) and the demand x for a particular clock radio are related by the equation x = 5000 - 50p. (a) express the price p in terms of the demand x, and find the domain of this function. (b) find the revenue r(x) from the sale of x clock radios. what is the domain of r? (c) find the marginal revenue at a production level of 3900 clock radios. (d) interpret r(4400)= - 76.00. what is the domain of r(x)? a. 0≤x≤50 b. x≤6000 c. x≥0 d. 0≤x≤5000 (c) the marginal revenue at a production level of 3900 clock radios is $ - 56. (round to the nearest cent as needed.) (d) which of the following statements is not a correct interpretation of r(4400)= - 76.00? a. if 4400 clock radios are produced, there will be no revenue, since r(4400) is negative. b. the total revenue for producing 4401 clock radios is approximately $76.00 less than the total revenue for producing 4400 clock radios. c. at a production level of 4400 clock radios, the change in revenue per unit change in production is approximately - 76.00 dollars.

the price p (in dollars) and the demand x for a particular clock radio are related by the equation x = 5000 - 50p. (a) express the price p in terms of the demand x, and find the domain of this function. (b) find the revenue r(x) from the sale of x clock radios. what is the domain of r? (c) find the marginal revenue at a production level of 3900 clock radios. (d) interpret r(4400)= - 76.00. what is the domain of r(x)? a. 0≤x≤50 b. x≤6000 c. x≥0 d. 0≤x≤5000 (c) the marginal revenue at a production level of 3900 clock radios is $ - 56. (round to the nearest cent as needed.) (d) which of the following statements is not a correct interpretation of r(4400)= - 76.00? a. if 4400 clock radios are produced, there will be no revenue, since r(4400) is negative. b. the total revenue for producing 4401 clock radios is approximately $76.00 less than the total revenue for producing 4400 clock radios. c. at a production level of 4400 clock radios, the change in revenue per unit change in production is approximately - 76.00 dollars.

Answer

Explanation:

Step1: Analyze the demand - price relationship

Given $x = 5000-50p$, we can solve for $p$: $50p=5000 - x$, so $p=\frac{5000 - x}{50}=100-\frac{x}{50}$. Since $x\geq0$ (non - negative demand) and $p\geq0$, when $p = 0$, $x = 5000$. So the domain of $p(x)$ is $0\leq x\leq5000$.

Step2: Find the revenue function

Revenue $R(x)=p\times x$. Substitute $p = 100-\frac{x}{50}$ into the revenue formula, we get $R(x)=(100-\frac{x}{50})x=100x-\frac{x^{2}}{50}$. The domain of $R(x)$ is the same as the domain of $x$ in the demand - price relationship, which is $0\leq x\leq5000$ because $x$ (demand) cannot be negative and is bounded by the maximum value when $p = 0$.

Step3: Find the marginal revenue function

The marginal revenue function $R'(x)$ is the derivative of $R(x)$. Since $R(x)=100x-\frac{x^{2}}{50}$, then $R'(x)=100-\frac{2x}{50}=100-\frac{x}{25}$. When $x = 3900$, $R'(3900)=100-\frac{3900}{25}=100 - 156=- 56$.

Step4: Interpret the marginal revenue value

The marginal revenue $R'(a)$ represents the approximate change in revenue when the production level changes from $a$ to $a + 1$. For $R'(4400)=-76$, it means that at a production level of 4400 clock - radios, the change in revenue per unit change in production is approximately - 76 dollars, or the total revenue for producing 4401 clock - radios is approximately $76$ less than the total revenue for producing 4400 clock - radios.

Answer:

(A) $p = 100-\frac{x}{50}$, domain: $0\leq x\leq5000$ (B) $R(x)=100x-\frac{x^{2}}{50}$, domain: $0\leq x\leq5000$ (C) - 56 (D) A. If 4400 clock radios are produced, there will be no revenue, since $R'(4400)$ is negative. (This is incorrect. A negative marginal revenue just means the revenue is decreasing as production increases by one unit, not that there is no revenue at all.)