Questions: Suppose a company has its total cost for a product given by C(x)=200 x+10,000 dollars and its total revenue given by R(x)=250 x-0.01 x^2 dollars, where x is the number of units produced and sold. Find the marginal revenue function R'(x)=250-0.02 x Find " R(72)-R(71) " and " R' (72) " and tell what each represents or predicts. Find the profit function and marginal profit function. P'(x) 50-0.02 x At which point does the profit reach its maximum? and the maximum profit? 52500

Suppose a company has its total cost for a product given by C(x)=200 x+10,000 dollars and its total revenue given by R(x)=250 x-0.01 x^2 dollars, where x is the number of units produced and sold.
Find the marginal revenue function
R'(x)=250-0.02 x
Find " R(72)-R(71) " and " R' (72) " and tell what each represents or predicts.
Find the profit function and marginal profit function.
P'(x) 50-0.02 x
At which point does the profit reach its maximum? and the maximum profit?
52500

Transcript text: Suppose a company has its total cost for a product given by $C(x)=200 x+10,000$ dollars and its total revenue given by $R(x)=250 x-0.01 x^{2}$ dollars, where x is the number of units produced and sold. Find the marginal revenue function \[ R^{\prime}(x)=250-0.02 x \] Find " $R(72)-R(71)$ " and " $R$ ' $(72)$ " and tell what each represents or predicts. Find the profit function and marginal profit function. \[ P^{\prime}(x) \quad 50-0.02 x \] At which point dose the profit reach its maximum? and the maximum profit? 52500

Solution

Solution Steps

Solution Approach

10-1. To find the marginal revenue function, we need to take the derivative of the revenue function $ R(x) = 250x - 0.01x^2 $ with respect to $ x $.

10-2. To find $ R(72) - R(71) $, we calculate the revenue at 72 units and subtract the revenue at 71 units. $ R'(72) $ is the derivative of the revenue function evaluated at $ x = 72 $, which represents the approximate change in revenue for an additional unit sold at 72 units.

10-3. The profit function $ P(x) $ is the difference between the revenue function $ R(x) $ and the cost function $ C(x) $. The marginal profit function is the derivative of the profit function with respect to $ x $.

Step 1: Marginal Revenue Function

To find the marginal revenue function, we differentiate the revenue function $ R(x) = -0.01x^2 + 250x $ with respect to $ x $: \[ R'(x) = 250 - 0.02x \]

Step 2: Calculate $ R(72) - R(71) $ and $ R'(72) $

Next, we compute the revenue at $ x = 72 $ and $ x = 71 $: \[ R(72) \approx 17948.16, \quad R(71) \approx 17699.59 \] Thus, the difference in revenue is: \[ R(72) - R(71) \approx 248.57 \] We also find the marginal revenue at $ x = 72 $: \[ R'(72) \approx 248.56 \]

Step 3: Profit Function and Marginal Profit Function

The profit function $ P(x) $ is given by the difference between revenue and cost: \[ P(x) = R(x) - C(x) = (-0.01x^2 + 250x) - (200x + 10000) = -0.01x^2 + 50x - 10000 \] The marginal profit function is the derivative of the profit function: \[ P'(x) = 50 - 0.02x \]