To solve [tex]\(\left(\frac{f}{g}\right)(3)\)[/tex] and interpret the answer, let's follow these detailed steps:
1. Evaluate [tex]\(f(3)\)[/tex]:
We start by calculating [tex]\(f(x)\)[/tex] when [tex]\(x = 3\)[/tex]. The function [tex]\(f(x)\)[/tex] is given by:
[tex]\[
f(x) = 6x + 8
\][/tex]
Plugging in [tex]\(x = 3\)[/tex]:
[tex]\[
f(3) = 6(3) + 8 = 18 + 8 = 26
\][/tex]
2. Evaluate [tex]\(g(3)\)[/tex]:
Next, we calculate [tex]\(g(x)\)[/tex] when [tex]\(x = 3\)[/tex]. The function [tex]\(g(x)\)[/tex] is given by:
[tex]\[
g(x) = x - 2
\][/tex]
Plugging in [tex]\(x = 3\)[/tex]:
[tex]\[
g(3) = 3 - 2 = 1
\][/tex]
3. Calculate [tex]\(\left(\frac{f}{g}\right)(3)\)[/tex]:
Now we find the value of [tex]\(\frac{f(3)}{g(3)}\)[/tex]:
[tex]\[
\left(\frac{f}{g}\right)(3) = \frac{f(3)}{g(3)} = \frac{26}{1} = 26
\][/tex]
4. Interpret the answer:
The result of [tex]\(\left(\frac{f}{g}\right)(3) = 26\)[/tex] represents the rate at which the cheetah is running. Since [tex]\(f(x)\)[/tex] represents the distance run by the cheetah in miles and [tex]\(g(x)\)[/tex] represents the time in hours, the value [tex]\(26\)[/tex] is the cheetah's speed in miles per hour.
Therefore, the correct interpretation is:
- 26; the cheetah's rate in miles per hour
So, the answer is:
[tex]\[ \boxed{26; \text{the cheetah's rate in miles per hour}} \][/tex]
