Final answer:
To determine the length of the supporting wire attached to the utility pole, we use the cosine function of the given 20° angle and the vertical distance of 37 ft from the top of the pole to the ground. The computed length of the wire is approximately 39.37 ft after rounding to hundredths.
Step-by-step explanation:
The question involves finding the length of a wire supporting a utility pole. Given the height of the pole as 40 ft and that the wire should be attached 3 ft from the top, this leaves us with a vertical distance of 40 ft - 3 ft = 37 ft. To find the length of the wire, we will use the trigonometric function cosine since the wire forms a 20° angle with the ground.
The wire, the ground, and the pole form a right-angled triangle. By using the cosine function:
cos(20°) = adjacent/hypotenuse
where the adjacent side is the vertical distance from the ground to the point of attachment (37 ft) and the hypotenuse is the length of the wire we want to find. Thus:
cos(20°) = 37 ft / wire length
Solving for the wire length:
wire length = 37 ft / cos(20°)
By calculating this, we find that the length of the wire is approximately:
wire length ≈ 37 ft / 0.93969 ≈ 39.37 ft
Therefore, the wire should be approximately 39.37 ft long, rounded to two decimal places.