Let's take a detailed step-by-step approach to solve this problem:
1. Given Information:
- [tex]\(\sin \theta = \frac{1}{2}\)[/tex]
- The equation [tex]\(\tan \theta = \frac{s^2}{49}\)[/tex]
2. Find [tex]\(\cos \theta\)[/tex]:
We know the identity involving sine and cosine:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Substituting the given value of [tex]\(\sin \theta = \frac{1}{2}\)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^2 + \cos^2 \theta = 1
\][/tex]
[tex]\[
\frac{1}{4} + \cos^2 \theta = 1
\][/tex]
[tex]\[
\cos^2 \theta = 1 - \frac{1}{4}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{3}{4}
\][/tex]
Taking the positive square root (since [tex]\(\cos\)[/tex] is positive in the first and fourth quadrants where sine is also positive):
[tex]\[
\cos \theta = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \approx 0.866
\][/tex]
3. Find [tex]\(\tan \theta\)[/tex]:
We use the identity for tangent:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
Substituting the values of [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \approx 0.577
\][/tex]
4. Substitute [tex]\(\tan \theta\)[/tex] into the equation to find [tex]\(s\)[/tex]:
The given equation is:
[tex]\[
\tan \theta = \frac{s^2}{49}
\][/tex]
Substituting the value of [tex]\(\tan \theta\)[/tex]:
[tex]\[
0.577 \approx \frac{s^2}{49}
\][/tex]
Solving for [tex]\(s^2\)[/tex]:
[tex]\[
s^2 = 0.577 \times 49 = 28.3
\][/tex]
Then, taking the square root of both sides gives:
[tex]\[
s = \sqrt{28.3} \approx 5.318
\][/tex]
5. Final Approximate Value:
The speed [tex]\(s\)[/tex] of the car in feet per second is approximately:
[tex]\[
s \approx 5.3
\][/tex]
So, the correct answer from the given options is:
[tex]\[ \boxed{5.3} \][/tex]
