Question

use the information provided to answer part a and part b for question 20.

1. a spring is attached at one end to support b and at the other end to collar a, as represented in the figure. collar a slides along the vertical bar between points c and d. in the figure, the angle θ is the angle created as the collar moves between points c and d.

part a
when θ = 28°, what is the distance from point a to point b to the nearest tenth of a foot?
enter your answer in the box.
(feet)

Explanation:

Step1: Identify the trigonometric relationship

We know that in a right - triangle (assuming the vertical line from B to the horizontal line through A and the line segment BA form a right - triangle with the vertical side as the adjacent side to the angle \(\theta\) and BA as the hypotenuse). The formula for the adjacent side \(x\) in a right - triangle is \(x = h\cos\theta\), where \(h\) is the length of the hypotenuse. Here, \(h = 3\) ft and \(\theta=28^{\circ}\).

Step2: Calculate the distance

We use the cosine function. \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so the distance \(d\) from point A to point B (adjacent side) is \(d = 3\cos(28^{\circ})\).
We know that \(\cos(28^{\circ})\approx0.8829\). Then \(d = 3\times0.8829 = 2.6487\approx2.6\)

Answer:

\(2.6\)