Question

3. For the years of 1983 to 1990, the number of mountain bike owners m (in millions) in the U.S. can be approximated by the model 푚(푡) = 0.337푡 2 −2.265푡 +3.962 where t = 3 represents 1983. a. In which year did 2.5 million people own mountain bikes? b. In what year was the number of mountain bike owners at a minimum?
© Success Academy Charter Schools 4 c. What was the maximum number of mountain bike owners between 1983 and 1990?

Answer 1

We can solve the given problem using the quadratic model \( m(t) = 0.337t^2 - 2.265t + 3.962 \), where \( t = 3 \) corresponds to the year 1983.

### a. In which year did 2.5 million people own mountain bikes?

To find when 2.5 million people owned mountain bikes, we set the model equal to 2.5:

\[
0.337t^2 - 2.265t + 3.962 = 2.5
\]

Subtracting 2.5 from both sides gives:

\[
0.337t^2 - 2.265t + 1.462 = 0
\]

Now we can use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 0.337 \), \( b = -2.265 \), and \( c = 1.462 \).

Calculating the discriminant:

\[
b^2 - 4ac = (-2.265)^2 - 4 \times 0.337 \times 1.462
\]

Calculating:

\[
(-2.265)^2 = 5.134225
\]
\[
4 \times 0.337 \times 1.462 = 1.975416
\]
\[
b^2 - 4ac = 5.134225 - 1.975416 = 3.158809
\]

Now, using the quadratic formula:

\[
t = \frac{-(-2.265) \pm \sqrt{3.158809}}{2 \times 0.337}
\]
\[
t = \frac{2.265 \pm 1.78}{0.674}
\]

Calculating the two potential solutions:

1. \( t = \frac{2.265 + 1.78}{0.674} \approx \frac{4.045}{0.674} \approx 6.006 \)
2. \( t = \frac{2.265 - 1.78}{0.674} \approx \frac{0.485}{0.674} \approx 0.719 \)

Rounding these values, we find:

- \( t \approx 6.006 \Rightarrow 1989 \)
- \( t \approx 0.719 \Rightarrow 1983 \)

Thus, **2.5 million people owned mountain bikes in 1983 and in 1989**.

### b. In what year was the number of mountain bike owners at a minimum?

The minimum value of a quadratic function occurs at the vertex. The vertex \( t \) for a quadratic \( ax^2 + bx + c \) can be calculated using:

\[
t = -\frac{b}{2a}
\]

Substituting our values:

\[
t = -\frac{-2.265}{2 \times 0.337} = \frac{2.265}{0.674} \approx 3.36
\]

Since \( t = 3 \) represents 1983, adding approximately 0.36 gives us about 0.36 years after 1983, which corresponds to roughly early 1984.

Thus, the minimum number of mountain bike owners occurred in **early 1984**.

### c. What was the maximum number of mountain bike owners between 1983 and 1990?

To find the maximum number of mountain bike owners, we can evaluate the function at the endpoints \( t = 3 \) (1983) and \( t = 7 \) (1990) as well as at the vertex \( t \approx 3.36 \).

Calculating \( m(3) \):

\[
m(3) = 0.337(3^2) - 2.265(3) + 3.962 = 0.337(9) - 6.795 + 3.962 = 3.033 - 6.795 + 3.962 \approx 0.200 million
\]

Calculating \( m(7) \):

\[
m(7) = 0.337(7^2) - 2.265(7) + 3.962 = 0.337(49) - 15.855 + 3.962 \approx 16.513 - 15.855 + 3.962 \approx 4.62 million
\]

Calculating \( m(3.36) \):

\[
m(3.36) = 0.337(3.36^2) - 2.265(3.36) + 3.962
\]

Calculating \( (3.36)^2 = 11.2976 \):

\[
m(3.36) = 0.337(11.2976) - 7.593 - 2.265 + 3.962
\]
\[
\approx 3.797 - 7.593 + 3.962 \approx 0.166 million
\]

Now let's summarize the maximum values:

- \( m(3) \approx 0.200 \) million
- \( m(7) \approx 4.62 \) million
- \( m(3.36) \approx 0.166 \) million

Thus, the maximum number of mountain bike owners between 1983 and 1990 was around **4.62 million** in **1990**.

### Summary
- **a.** 1983 and 1989
- **b.** Early 1984
- **c.** 4.62 million in 1990