Given the problem, we are asked to find the speed [tex]\( s \)[/tex] of a toy car in feet per second using the given information about the angle of incline [tex]\( \theta \)[/tex], where [tex]\(\sin \theta = \frac{1}{2}\)[/tex], and the relationship [tex]\(\tan \theta = \frac{s^2}{49}\)[/tex].
Let's solve this step-by-step:
1. Given:
[tex]\[
\sin \theta = \frac{1}{2}
\][/tex]
2. Using the Pythagorean Identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Substitute [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]
[tex]\[
\cos \theta = \sqrt{\frac{3}{4}}
\][/tex]
[tex]\[
\cos \theta = \frac{\sqrt{3}}{2}
\][/tex]
3. Calculate [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
[tex]\[
\tan \theta = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}
\][/tex]
[tex]\[
\tan \theta = \frac{1}{\sqrt{3}}
\][/tex]
[tex]\[
\tan \theta = \frac{\sqrt{3}}{3}
\][/tex]
4. Substitute [tex]\(\tan \theta\)[/tex] into the equation:
[tex]\[
\tan \theta = \frac{s^2}{49}
\][/tex]
[tex]\[
\frac{\sqrt{3}}{3} = \frac{s^2}{49}
\][/tex]
5. Solve for [tex]\(s^2\)[/tex]:
[tex]\[
s^2 = 49 \cdot \frac{\sqrt{3}}{3}
\][/tex]
[tex]\[
s^2 = 49 \cdot \frac{\sqrt{3}}{3}
\][/tex]
6. Find [tex]\(s\)[/tex] by taking the square root:
[tex]\[
s = \sqrt{49 \cdot \frac{\sqrt{3}}{3}}
\][/tex]
[tex]\[
s = \frac{\sqrt{49 \cdot \sqrt{3}}}{\sqrt{3}}
\][/tex]
7. Approximate the value of [tex]\(s\)[/tex]:
After evaluating and approximating, we find:
[tex]\[
s \approx 5.3
\][/tex]
Thus, the approximate value of [tex]\( s \)[/tex] is [tex]\( \boxed{5.3} \)[/tex].
Answer: 5.3
