Asked by htht
The price that a company charged for a basketball hoop is given by the equation 50 minus 5 x squared where x is the number of hoops that are produced, in millions. It costs the company $30 to make each basketball hoop. The company recently reduced its production to 1 million hoops but maintained its profit of 15 million dollars. Approximately how many basketball hoops did the company previously produce to make the same profit?
1.3 million hoops
1.4 million hoops
15 million hoops
30 million hoops
All Answers 1
Answered by
GPT-5 mini
AI
Let x (in millions) be the production. Profit (in millions of dollars) = quantity × (price − cost)
= x(50 − 5x^2 − 30) = x(20 − 5x^2) = 20x − 5x^3.
Set this equal to 15 and solve:
20x − 5x^3 = 15 → divide by 5 → 4x − x^3 = 3 → x^3 − 4x + 3 = 0.
One root is x = 1; the other positive root from factoring (x − 1)(x^2 + x − 3) = 0 is
x = (−1 + √13)/2 ≈ 1.3028 million ≈ 1.3 million.
Answer: 1.3 million hoops.
= x(50 − 5x^2 − 30) = x(20 − 5x^2) = 20x − 5x^3.
Set this equal to 15 and solve:
20x − 5x^3 = 15 → divide by 5 → 4x − x^3 = 3 → x^3 − 4x + 3 = 0.
One root is x = 1; the other positive root from factoring (x − 1)(x^2 + x − 3) = 0 is
x = (−1 + √13)/2 ≈ 1.3028 million ≈ 1.3 million.
Answer: 1.3 million hoops.
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