A guy wire runs from the ground to a cell tower. The wire is attached to the cell tower 150 feet above the ground. The angle formed between the wire and the ground is 40° (see figure). (Round your answers to one decimal place.)

We form a rt triangle:

Y = 150 Ft. = Ver. side.
X = Hor. side = Dist. from bottom of tower to point where wire attaches to gnd,
Z = hyp. = Length of wire.

Z = Y/sin40 = 150 / sin40 = 233.4 Ft.
X = Z*cos40 = 233.4*cos40178.8 Ft.

To solve this problem, we can use trigonometry. Specifically, we can use the concept of the sine function to find the length of the guy wire.

Let's label the height of the cell tower as "h" (150 feet) and the angle formed between the wire and the ground as "θ" (40°).

We know that the sine of an angle is equal to the opposite side divided by the hypotenuse in a right triangle. In this case, the height of the cell tower is the opposite side and the length of the guy wire is the hypotenuse.

Using the sine function:
sin(θ) = opposite/hypotenuse

Substituting the given values:
sin(40°) = h/hypotenuse

Now, we can solve for the hypotenuse, which represents the length of the guy wire:
hypotenuse = h/sin(θ)

Plugging in the values:
hypotenuse = 150 / sin(40°)

Using a scientific calculator or trigonometric table, we can find the sine of 40°:
sin(40°) ≈ 0.6428

Now, we can substitute this value back into the equation:
hypotenuse = 150 / 0.6428

Calculating this expression, we find:
hypotenuse ≈ 233.4 feet

Therefore, the length of the guy wire is approximately 233.4 feet.

To solve this problem, we can use trigonometric ratios to find the length of the guy wire and the distance between its attachment point and the base of the cell tower.

Step 1: Identify the given information:
- The height of the cell tower where the wire is attached is 150 feet.
- The angle formed between the guy wire and the ground is 40°.

Step 2: Determine the trigonometric ratio to use:
In this case, we know the angle and the opposite side (height of the cell tower). Therefore, we can use the tangent ratio.

Step 3: Apply the tangent ratio:
The tangent of an angle is equal to the ratio of the length of the opposite side to the adjacent side.
In this case, the opposite side is the height of the cell tower (150 ft), and the adjacent side is unknown (let's call it x ft):

tan(40°) = opposite/adjacent
tan(40°) = 150/x

Step 4: Solve for x:
To isolate x, we can cross multiply the equation and solve for x:

x * tan(40°) = 150
x = 150/tan(40°)

Using a calculator, we can find that tan(40°) ≈ 0.8391.

Therefore, x ≈ 150/0.8391 ≈ 178.8 ft.

So the distance between the attachment point of the guy wire and the base of the cell tower is approximately 178.8 feet.