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
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Solution

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Solution Steps

Solution Approach

10-1. To find the marginal revenue function, we need to take the derivative of the revenue function R(x)=250x0.01x2 R(x) = 250x - 0.01x^2 with respect to x x .

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

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

Step 1: Marginal Revenue Function

To find the marginal revenue function, we differentiate the revenue function R(x)=0.01x2+250x R(x) = -0.01x^2 + 250x with respect to x x : R(x)=2500.02x R'(x) = 250 - 0.02x

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

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

Step 3: Profit Function and Marginal Profit Function

The profit function P(x) P(x) is given by the difference between revenue and cost: P(x)=R(x)C(x)=(0.01x2+250x)(200x+10000)=0.01x2+50x10000 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)=500.02x P'(x) = 50 - 0.02x

Final Answer

  • Marginal Revenue Function: R(x)=2500.02x R'(x) = 250 - 0.02x
  • R(72)R(71)248.57 R(72) - R(71) \approx 248.57
  • R(72)248.56 R'(72) \approx 248.56
  • Profit Function: P(x)=0.01x2+50x10000 P(x) = -0.01x^2 + 50x - 10000
  • Marginal Profit Function: P(x)=500.02x P'(x) = 50 - 0.02x

R(x)=2500.02x,R(72)R(71)248.57,R(72)248.56 \boxed{R'(x) = 250 - 0.02x, \quad R(72) - R(71) \approx 248.57, \quad R'(72) \approx 248.56}

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