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A student uses the equation [tex]\tan \theta = \frac{s^2}{49}[/tex] to represent the speed, [tex]s[/tex], in feet per second, of a toy car driving around a circular track having an angle of incline [tex]\theta[/tex], where [tex]\sin \theta = \frac{1}{2}[/tex]. To solve the problem, the student used the given value of [tex]\sin \theta[/tex] to find the value of [tex]\tan \theta[/tex] and then substituted the value of [tex]\tan \theta[/tex] in the equation above to solve for [tex]s[/tex].

What is the approximate value of [tex]s[/tex], the speed of the car in feet per second?

A. 5.3
B. 7.5
C. 9.2
D. 28.3

Sagot :

GinnyAnswer
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
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