To determine the approximate value of [tex]\( s \)[/tex], the speed of the car in feet per second, we can follow a detailed step-by-step process based on the given conditions.
1. Given Information:
- The equation to find the speed is [tex]\(\tan \theta = \frac{s^2}{49}\)[/tex].
- The sine of the angle [tex]\(\theta\)[/tex] is given as [tex]\(\sin \theta = \frac{1}{2}\)[/tex].
2. Step 1: Calculate [tex]\(\theta\)[/tex]:
We know that [tex]\(\sin \theta = \frac{1}{2}\)[/tex]. Using the inverse sine function (arcsine), we find:
[tex]\[
\theta = \arcsin \left( \frac{1}{2} \right)
\][/tex]
The value of [tex]\(\theta\)[/tex] in radians is approximately [tex]\(0.5236\)[/tex].
3. Step 2: Calculate [tex]\(\tan \theta\)[/tex]:
Using the calculated value of [tex]\(\theta\)[/tex], we now find [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan(0.5236) \approx 0.5774
\][/tex]
4. Step 3: Substitute [tex]\(\tan \theta\)[/tex] into the equation [tex]\(\tan \theta = \frac{s^2}{49}\)[/tex]:
We substitute [tex]\(0.5774\)[/tex] into the equation:
[tex]\[
0.5774 = \frac{s^2}{49}
\][/tex]
5. Step 4: Solve for [tex]\(s^2\)[/tex]:
Rearrange the equation to solve for [tex]\(s^2\)[/tex]:
[tex]\[
s^2 = 49 \times 0.5774 \approx 28.2902
\][/tex]
6. Step 5: Solve for [tex]\(s\)[/tex]:
Take the square root of [tex]\(28.2902\)[/tex] to find [tex]\(s\)[/tex]:
[tex]\[
s = \sqrt{28.2902} \approx 5.3188
\][/tex]
Therefore, the approximate value of [tex]\( s \)[/tex], the speed of the car in feet per second, is closest to [tex]\( 5.3 \)[/tex]. Hence, the correct answer is:
[tex]\[
\boxed{5.3}
\][/tex]
