Select the correct answer.A manufacturing company plans to progressively increase its production capacity over the next few quarters. (A quarter is a period of three months.) The increase in production can be modeled by the equation y = x6 − 25x4 + 199x2, where x is the number of quarters. What is the minimum duration required for the company to reach a production capacity of 4,975 units? A. 4 quarters B. 5 quarters C. 6 quarters D. 7 quarters
Question
Select the correct answer.A manufacturing company plans to progressively increase its production capacity over the next few quarters. (A quarter is a period of three months.) The increase in production can be modeled by the equation y = x6 − 25x4 + 199x2, where x is the number of quarters. What is the minimum duration required for the company to reach a production capacity of 4,975 units? A. 4 quarters B. 5 quarters C. 6 quarters D. 7 quarters
Solution
To solve this problem, we need to set the equation y = x^6 - 25x^4 + 199x^2 equal to 4,975 and solve for x.
y = x^6 - 25x^4 + 199x^2 = 4975
This is a complex polynomial equation. Solving it exactly would require advanced mathematical techniques. However, we can estimate the solution by trying different values of x (number of quarters) and see which one gives a result closest to 4975.
Let's try the options given:
A. 4 quarters: y = 4^6 - 254^4 + 1994^2 = 4096 - 1600 + 3196 = 5692 B. 5 quarters: y = 5^6 - 255^4 + 1995^2 = 15625 - 3125 + 4975 = 17475 C. 6 quarters: y = 6^6 - 256^4 + 1996^2 = 46656 - 21600 + 7182 = 32238 D. 7 quarters: y = 7^6 - 257^4 + 1997^2 = 117649 - 8575 + 9603 = 118677
As we can see, the value of y is closest to 4975 when x = 4. Therefore, the minimum duration required for the company to reach a production capacity of 4,975 units is 4 quarters. So, the correct answer is A. 4 quarters.
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