Question

6 formula 0.5 points
given right triangle abc, where side \c\ is the hypotenuse, angle a measures 60 degrees, and side b measures 16 m, find the length of side a.
be sure your calculator is in deg mode, and use the proper trig function on your calculator in the computation.
round your answer to the nearest hundredth.
answer

7 formula 0.5 points
a 31-foot long wire will be attached to the top of a pole for support, and will be pulled tight and anchored to the ground.
if the wire makes a 18 degree angle with the ground, how tall is the pole?
image: diagram with wire, pole, and ground
be sure your calculator is in deg mode, and use the proper trig function on your calculator in the computation.
round your answer to the nearest tenth of a foot, but do not include \ft\ with your response.

Explanation:

Question 6

Step1: Identify trigonometric ratio

In right triangle \(ABC\), \(\tan(A)=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}\). Given \(A = 60^\circ\) and \(b = 16\) m.

Step2: Solve for \(a\)

Rearrange the formula: \(a=b\times\tan(A)\). Substitute \(b = 16\) and \(A = 60^\circ\). So \(a = 16\times\tan(60^\circ)\). \(\tan(60^\circ)=\sqrt{3}\approx1.732\). Then \(a = 16\times1.732 = 27.712\approx27.71\) (rounded to nearest hundredth).

Step1: Identify trigonometric ratio

The wire, pole, and ground form a right triangle. The wire is the hypotenuse (\(c = 31\) ft), the pole is the opposite side to the angle \(18^\circ\), let the height of the pole be \(h\). So \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{h}{c}\).

Step2: Solve for \(h\)

Rearrange: \(h = c\times\sin(\theta)\). Substitute \(c = 31\) and \(\theta = 18^\circ\). So \(h = 31\times\sin(18^\circ)\). \(\sin(18^\circ)\approx0.3090\). Then \(h = 31\times0.3090 = 9.579\approx9.6\) (rounded to nearest tenth).

Answer:

\(27.71\)

Question 7