The total cost (in dollars) for a company to manufacture and sell 𝑥 items per week is 𝐶=50𝑥+2960, whereas the revenue brought in by selling all 𝑥 items is 𝑅=140𝑥-0.6𝑥2. How many items must be sold to obtain a weekly profit of $400?Hint: Profit = Revenue - Cost.They need to sell or items.
Question
The total cost (in dollars) for a company to manufacture and sell 𝑥 items per week is 𝐶=50𝑥+2960, whereas the revenue brought in by selling all 𝑥 items is 𝑅=140𝑥-0.6𝑥2. How many items must be sold to obtain a weekly profit of $400?Hint: Profit = Revenue - Cost.They need to sell or items.
Solution
To find out how many items must be sold to obtain a weekly profit of 400 and solve for x.
Given: Cost, C = 50x + 2960 Revenue, R = 140x - 0.6x^2 Profit = R - C
We substitute the given equations into the profit equation:
Profit = (140x - 0.6x^2) - (50x + 2960)
Simplify the equation:
Profit = 90x - 0.6x^2 - 2960
We want to find the number of items (x) that will give a profit of $400, so we set the equation equal to 400 and solve for x:
400 = 90x - 0.6x^2 - 2960
Rearrange the equation:
0.6x^2 - 90x + 3360 = 0
This is a quadratic equation in the form ax^2 + bx + c = 0. We can solve for x using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a:
x = [90 ± sqrt((90)^2 - 40.63360)] / (2*0.6)
Solving this equation will give us the number of items that need to be sold to obtain a weekly profit of $400.
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