A company manufactures and sells electronic gadgets. The company's revenue, [tex]$R$[/tex], in thousands of dollars, can be modeled by the function [tex]$R(x)=-2 x^2+100 x$[/tex], where [tex]$x$[/tex] represents the number of gadgets sold in thousands. Find the answers algebraically, ensuring that the units are expressed correctly. Remember to show your work for full credit.
a. The company wants to know the ideal sales target to generate the highest possible revenue. How many gadgets should they aim to sell?
b. What is the maximum possible revenue the company can achieve?
c. What would be the revenue if the company sells 10,000 gadgets?
d. At what number of gadgets sold does the company have zero revenue?
Answer (2)
Find the vertex of the quadratic revenue function R ( x ) = − 2 x 2 + 100 x to determine the sales target for maximum revenue: x = 2 a − b = 25 .
Calculate the maximum revenue by substituting x = 25 into the revenue function: R ( 25 ) = 1250 .
Determine the revenue at x = 10 : R ( 10 ) = 800 .
Solve for x when R ( x ) = 0 to find the sales targets for zero revenue: x = 0 and x = 50 . The final answers are:
a. 25000 gadgets
b. 1250000 dollars
c. 800000 dollars
d. 0 or 50000 gadgets
Explanation
Understanding the Revenue Function The problem provides a revenue function R ( x ) = − 2 x 2 + 100 x , where R is the revenue in thousands of dollars and x is the number of gadgets sold in thousands. We need to find the sales target to maximize revenue, the maximum revenue, the revenue at a specific sales target, and the sales target for zero revenue.
Finding the Ideal Sales Target To find the ideal sales target to maximize revenue, we need to find the vertex of the parabola represented by the revenue function. The x-coordinate of the vertex is given by the formula x = 2 a − b , where a = − 2 and b = 100 . Plugging in these values, we get: x = 2 ( − 2 ) − 100 = − 4 − 100 = 25 This means the company should aim to sell 25,000 gadgets to maximize revenue.
Calculating the Maximum Revenue To find the maximum possible revenue, we substitute the x-value we found in the previous step ( x = 25 ) into the revenue function: R ( 25 ) = − 2 ( 25 ) 2 + 100 ( 25 ) = − 2 ( 625 ) + 2500 = − 1250 + 2500 = 1250 So, the maximum possible revenue the company can achieve is $1,250,000.
Determining Revenue at 10,000 Gadgets To find the revenue when the company sells 10,000 gadgets, we substitute x = 10 into the revenue function: R ( 10 ) = − 2 ( 10 ) 2 + 100 ( 10 ) = − 2 ( 100 ) + 1000 = − 200 + 1000 = 800 Therefore, the revenue would be $800,000 if the company sells 10,000 gadgets.
Finding the Sales Target for Zero Revenue To find the number of gadgets sold when the revenue is zero, we need to solve the equation R ( x ) = − 2 x 2 + 100 x = 0 for x . We can factor out an x from the equation: x ( − 2 x + 100 ) = 0 This gives us two possible solutions:
x = 0
− 2 x + 100 = 0 ⇒ 2 x = 100 ⇒ x = 50 So, the company has zero revenue when they sell 0 gadgets or 50,000 gadgets.
Final Answers In summary: a. The company should aim to sell 25,000 gadgets to maximize revenue. b. The maximum possible revenue is $1,250,000. c. The revenue if the company sells 10,000 gadgets is $800,000. d. The company has zero revenue when they sell 0 or 50,000 gadgets.
Examples
Understanding revenue functions is crucial for businesses. For example, a local bakery can use a similar quadratic function to model their daily profit based on the number of cakes sold. By finding the vertex of this function, they can determine the optimal number of cakes to bake each day to maximize their profit. This helps in inventory management and resource allocation, ensuring they don't overproduce or underproduce, leading to efficient operations and higher profitability.
The company should aim to sell 25,000 gadgets to achieve maximum revenue of $1,250,000. Selling 10,000 gadgets would yield a revenue of $800,000, and the company would earn zero revenue by selling either 0 or 50,000 gadgets.
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