Question

in the triangle shown above, if \\( \theta \\) increases at a constant rate of 3 radians per minute, at what rate is \\( x \\) increasing in units per minute when \\( x \\) equals 3 units?
a 3
b \\( \frac { 15 } { 4 } \\)
c 4
d 9
e 12

Explanation:

Step1: Establish the relationship between \(x\) and \(\theta\)

Using the sine function in a right - triangle: \(\sin\theta=\frac{x}{5}\), so \(x = 5\sin\theta\).

Step2: Differentiate both sides with respect to time \(t\)

By the chain rule, \(\frac{dx}{dt}=5\cos\theta\frac{d\theta}{dt}\).

Step3: Find \(\cos\theta\) when \(x = 3\)

Since \(x = 5\sin\theta\) and \(x = 3\), then \(\sin\theta=\frac{3}{5}\). Using the identity \(\sin^{2}\theta+\cos^{2}\theta = 1\), we have \(\cos\theta=\sqrt{1-\sin^{2}\theta}=\sqrt{1 - (\frac{3}{5})^{2}}=\frac{4}{5}\).

Step4: Substitute \(\cos\theta\) and \(\frac{d\theta}{dt}\) into the derivative equation

We know that \(\frac{d\theta}{dt}=3\) radians per minute. Substituting \(\cos\theta=\frac{4}{5}\) and \(\frac{d\theta}{dt}=3\) into \(\frac{dx}{dt}=5\cos\theta\frac{d\theta}{dt}\), we get \(\frac{dx}{dt}=5\times\frac{4}{5}\times3\).

Answer:

\(12\) (Option E)