Question

The bar graph shows the number of fatal vehicle crashes per 100 million miles driven for drivers of various age groups. The number of fatal vehicle crashes per 100 million​ miles, N, for drivers of age x can be modeled by the following formula.
Upper N equals 0.013 x squared minus 1.18 x plus 28.23
A bar graph titled Age of Drivers and Fatal Crashes has a vertical axis labeled Fatalities per 100 Million Miles from 0 to 24 in increments of 3. The horizontal axis is labeled Age of Drivers. There are vertical bars with labels and heights as follows: 16, 12.7; 18, 11.2; 20, 9.8; 25, 6.9; 35, 2.9; 45, 1.5; 55, 2.7; 65, 6.5; 75, 12.9; 79, 16.1. Each bar is labeled with its height.
Age of Drivers and Fatal Crashes
16
18
20
25
35
45
55
65
75
79
0
6
12
18
24
Fatalities/100 Million Miles
font size decreased by 3 12.7font size decreased by 3 11.2font size decreased by 3 9.8font size decreased by 3 6.9font size decreased by 3 2.9font size decreased by 3 1.5font size decreased by 3 2.7font size decreased by 3 6.5font size decreased by 3 12.9font size decreased by 3 16.1
Age of Drivers
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Part 1
According to the​ formula, what age groups are expected to be involved in 5 fatal crashes per 100 million miles​ driven?
  
enter your response here ​year-olds.

Answer 1

To find the age groups that are expected to be involved in 5 fatal crashes per 100 million miles driven, we need to solve the equation given by the formula:

\[ N = 0.013x^2 - 1.18x + 28.23 \]

We set \( N = 5 \):

\[ 5 = 0.013x^2 - 1.18x + 28.23 \]

Rearranging gives us:

\[ 0.013x^2 - 1.18x + 28.23 - 5 = 0 \]

Which simplifies to:

\[ 0.013x^2 - 1.18x + 23.23 = 0 \]

Next, we can use the quadratic formula to solve for \( x \):

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 0.013 \), \( b = -1.18 \), and \( c = 23.23 \).

Calculating the discriminant:

\[ b^2 - 4ac = (-1.18)^2 - 4(0.013)(23.23) \]

Calculating each part:

1. \( (-1.18)^2 = 1.3924 \)
2. \( 4 \times 0.013 \times 23.23 = 1.20508 \)

Now we substitute that back into the discriminant:

\[ 1.3924 - 1.20508 = 0.18732 \]

Now, we can calculate \( x \):

\[ x = \frac{1.18 \pm \sqrt{0.18732}}{0.026} \]
\[ \sqrt{0.18732} \approx 0.4331 \]

Thus:

\[ x = \frac{1.18 \pm 0.4331}{0.026} \]

This gives us two potential solutions:

1. \( x = \frac{1.18 + 0.4331}{0.026} \)
2. \( x = \frac{1.18 - 0.4331}{0.026} \)

Calculating both:

1. \( x_1 \approx \frac{1.6131}{0.026} \approx 62.4 \) (approximately 62 years old)
2. \( x_2 \approx \frac{0.7469}{0.026} \approx 28.8 \) (approximately 29 years old)

Thus, the two age groups expected to be involved in 5 fatal crashes per 100 million miles driven are:

**29-year-olds and 62-year-olds.**