A hoverboard manufacturer has just announced the Glide 5 hoverboard. The accounting department has determined that the cost to manufacture the Glide 5 hoverboard and the equation is [tex]$y=22.68 x+28249$[/tex]. The revenue equation is [tex]$y=75.98 x$[/tex].
What is the break-even point for the Glide 5 hoverboard?
The break even point for the Glide 5 hoverboard is [$\square$].
(Enter your answer as an ordered pair. Round [tex]$x$[/tex] to the nearest whole number and [tex]$y$[/tex] to two decimal places.)
Answer (2)
Set the cost equation equal to the revenue equation: 22.68 x + 28249 = 75.98 x .
Solve for x : x = 53.3 28249 = 530 .
Substitute x = 530 into the revenue equation: y = 75.98 × 530 = 40269.4 .
The break-even point is ( 530 , 40269.40 ) .
Explanation
Understanding the Problem We are given the cost equation y = 22.68 x + 28249 and the revenue equation y = 75.98 x , where x represents the number of hoverboards and y represents the cost or revenue. Our goal is to find the break-even point, which is the point where the cost equals the revenue.
Setting up the Equation To find the break-even point, we need to set the cost equation equal to the revenue equation: 22.68 x + 28249 = 75.98 x
Isolating x Now, we solve for x by subtracting 22.68 x from both sides of the equation: 28249 = 75.98 x − 22.68 x
Simplifying Simplify the equation: 28249 = 53.3 x
Calculating x Divide both sides by 53.3 to find the value of x :
x = 53.3 28249 x = 530
Calculating y Since we are asked to round x to the nearest whole number, and x is already a whole number, we don't need to round it. Now we substitute x = 530 into the revenue equation to find the value of y :
y = 75.98 x = 75.98 × 530 = 40269.4
Finding the Break-Even Point We are asked to round y to two decimal places, so y = 40269.40 . Therefore, the break-even point is ( 530 , 40269.40 ) .
Examples
Understanding break-even points is crucial for businesses. For example, if a company is launching a new product, knowing the break-even point helps them determine how many units they need to sell to cover their production costs. This information is vital for setting sales targets, pricing strategies, and making informed decisions about the product's viability. By calculating the break-even point, businesses can assess the risk associated with the new product and make necessary adjustments to improve profitability. In our case, the hoverboard company knows they need to sell 530 Glide 5 hoverboards to start making a profit.
The break-even point for the Glide 5 hoverboard is (530, 40269.40), meaning the manufacturer needs to sell 530 units to cover the costs, generating revenue of approximately $40,269.40. This calculation helps the manufacturer understand the minimum sales needed to avoid losses.
;
