Question

a student uses the equation tanθ = s²/49 to represent the speed, s, in feet per second, of a toy car driving around a circular track having an angle of incline θ, where sinθ = 1/2. to solve the problem, the student used the given value of sinθ to find the value of tanθ and then substituted the value of tanθ in the equation above to solve for s. what is the approximate value of s, the speed of the car in feet per second? 5.3 7.5 9.2 28.3

Explanation:

Step1: Find $\cos\theta$

Given $\sin\theta=\frac{1}{2}$, using $\sin^{2}\theta+\cos^{2}\theta = 1$, we have $\cos\theta=\sqrt{1 - \sin^{2}\theta}=\sqrt{1-(\frac{1}{2})^{2}}=\sqrt{1-\frac{1}{4}}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$

Step2: Calculate $\tan\theta$

Since $\tan\theta=\frac{\sin\theta}{\cos\theta}$, substituting $\sin\theta=\frac{1}{2}$ and $\cos\theta=\frac{\sqrt{3}}{2}$, we get $\tan\theta=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}\approx0.577$

Step3: Solve for $s$

We know $\tan\theta=\frac{s^{2}}{49}$, substituting $\tan\theta\approx0.577$, we have $0.577=\frac{s^{2}}{49}$, then $s^{2}=49\times0.577 = 28.273$, and $s=\sqrt{28.273}\approx5.3$

Answer:

5.3