Answer :
The age at which the accident rate is a minimum is approximately 48.163 years. The minimum accident rate is approximately 73.797.
To find the age at which the accident rate is a minimum and the corresponding minimum rate, we can find the critical points of the function [tex]f(x) = 98.5 - 2.36x + 0.0245x^2[/tex] within the given interval.
First, let's find the derivative of the function f(x):
f'(x) = -2.36 + 0.049x
Next, we set f'(x) equal to zero and solve for x to find the critical point:
-2.36 + 0.049x = 0
0.049x = 2.36
x = 2.36 / 0.049
x ≈ 48.163
The critical point occurs at x ≈ 48.163.
To confirm whether this critical point is a minimum or maximum, we can analyze the second derivative:
f''(x) = 0.049
Since the second derivative is positive (0.049 > 0), the critical point represents a minimum.
Therefore, the age at which the accident rate is a minimum is approximately 48.163 years. To find the minimum rate, we substitute this value back into the function:
[tex]f(48.163) = 98.5 - 2.36(48.163) + 0.0245(48.163)^2[/tex]
Calculating this expression will give us the minimum rate.
[tex]f(48.163) = 98.5 - 2.36(48.163) + 0.0245(48.163)^2[/tex]
≈ 73.797
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