For years of 1980 to 1990, the number of mountain owners m ( in millions ) in the US can be approximated by the model m=.377t^2 – 2.265t + 3.962,3 ≤ t ≤ 10 where t = 3 represents 1983. 1. In which year did 2.5 million people own mountain bikes? 2. In what year was the number of mountain bike owners at a minimum? I'll really appreciated if it's was posible to explain how did you got the answer. Thank you

The answers to the question are

1. 1985

2. 1983

Step-by-step explanation:

To solve the question we note that the equation is

To find the the year in which 2.5 milion people own mountain bikes we have

to equate to 2.5 million therefore

= 2.5 or 0.377·t² - 2.265·t + 3.962-2.5 = 0

0.377·t² - 2.265·t + 1.462 = 0 which gives (t-0.736)×(t-5.272)×0.377 =  0

That is t = 0.736 or 5.272 and the range is 3 ≤ t ≤ 10

the answer is 5.272 year or

1985

2. For the minimum mountain bike owners we have

0.377·t² - 2.265·t + 3.962 = minimum when  0.377·t² - 2.265·t + 3.962 = 0

Therefore we have

0.377·t² - 2.265·t + 3.962 = 0 or Solving with the quadratic formula we have

(2.265±√(2.265²-4×0.377×3.962))÷(2×0.377) Which is indeterminate

hence we differentiate the expression to get and equate to zero to get the minimum

0.754·t -2.265 =0 which gives t = 2.265/0.754 = 3.003979

or the 3rd year. That is

1983