Video: US-SAT03S4-Q18-516124893207

A company produces lollipops. In the equations 𝐢(π‘₯) = 3000 + (3/2 π‘₯), 𝑅(π‘₯) = 2π‘₯, the function 𝐢(π‘₯) represents the total production cost, in dollars, of π‘₯ lollipops. And the function 𝑅(π‘₯) represents the total revenue, in dollars, from selling π‘₯ lollipops. How many lollipops should be produced so that the total cost of production is equal to the total revenue gained from selling them?

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Video Transcript

A company produces lollipops. In the equations 𝐢 of π‘₯ equals 3000 plus three-halves π‘₯, 𝑅 of π‘₯ equals two π‘₯, the function 𝐢 of π‘₯ represents the total production cost, in dollars, of π‘₯ lollipops. And the function 𝑅 of π‘₯ represents the total revenue, in dollars, from selling π‘₯ lollipops. How many lollipops should be produced so that the total cost of production is equal to the total revenue gained from selling them?

We know that 𝐢 of π‘₯ is our production cost. And 𝑅 of π‘₯ is our revenue. We’re looking for the place where these two values are equal. 3000 plus three-halves π‘₯ equals two π‘₯, production equal to revenue. At this point, we want to solve for π‘₯, which will tell us how many lollipops need to be produced so that these two equations are equal. First, subtract three-halves π‘₯ from both sides of the equation. I know that two is equal to four over two. And four over two minus three over two equals one-half. So we have one-half π‘₯ remaining. Two minus three-halves or two minus one and a half is equal to one-half. And we have the variable π‘₯.

Now, if one-half π‘₯ equals 3000, we can multiply both sides of the equation to find out how much π‘₯ is equal to. One-half π‘₯ times two equals π‘₯. And two times 3000 equals 6000. This means that when the company produces 6000 lollipops, the cost of production is equal to the revenue gained from selling the same amount. We can check that this is true. First with the production cost, 3000 plus three-halves times 6000 equals 12000 dollars. That’s how much it costs to produce 6000 lollipops. And we multiply two times 6000 to get the total revenue from selling 6000 lollipops. Two times 6000 is 12000, which confirms what we found.

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