A vertical pole 40 feet tall stands on a hillside that makes an angle of 17 degrees with the horizontal. Approximate the minimum length of cable that will reach the top of the pole from a point on the hillside 72 feet downhill from the base of the flagpole.
Hmmm. The 72 feet is measured on the slope, not out from the pole horizontally. I get
top of pole is at
40 + 72 sin17°
downhill point is
72 cos 17°
from point beneath the pole
cable^2 = (40+72 sin17°)^2 + (72 cos 17°)^2
vertical pole 20 feet high is sighted horizontally from a point 75 feet from its base up the sloping side site of a hill on which it rest.what is the inclination of the slope?
To approximate the minimum length of the cable needed to reach the top of the pole, we can use trigonometry.
First, let's draw a diagram to better understand the situation:
```
P
|\
| \
| \
| \
| \ 40 ft
| \
| \
| \
| -----
| -----
| 72 ft
|-----------------
B A
```
Here, P represents the top of the pole, B represents the base of the pole, and A represents the point on the hillside. We are looking for the minimum distance from A to P.
To find the distance from A to P, we can break it down into two components: the vertical distance from A to B (which is the height of the pole, 40 feet), and the horizontal distance from B to P.
Considering the right triangle formed by point A, B, and the foot of the pole, we can use the trigonometric function tangent to find the angle A (θ):
tan(θ) = Opposite / Adjacent
In this case, the opposite side is 40 feet (the height of the pole) and the adjacent side is 72 feet (the horizontal distance from B to A). Therefore:
tan(θ) = 40 / 72
To find θ, we take the inverse tangent (or arctan) of both sides:
θ = arctan (40 / 72)
Using a calculator, we find that θ ≈ 29.2 degrees.
Now we have two angles: 17 degrees (the angle of the hillside) and 29.2 degrees (θ).
Considering the right triangle formed by point P, B, and the foot of the pole, we can use the trigonometric function sine to find the ratio of opposite/adjacent sides:
sin(θ + 17) = Opposite / Hypotenuse
In this case, the opposite side is the height of the pole (40 feet), and we want to find the hypotenuse (which represents the minimum length of the cable). Therefore:
sin(θ + 17) = 40 / Hypotenuse
Rearranging the equation, we can solve for the hypotenuse:
Hypotenuse = 40 / sin(θ + 17)
Using a calculator, we find that sin(θ + 17) ≈ sin(46.2) ≈ 0.7210.
Substituting this value back into the equation, we find:
Hypotenuse ≈ 40 / 0.7210
Hypotenuse ≈ 55.4 feet
Therefore, the minimum length of the cable needed to reach the top of the pole from a point on the hillside 72 feet downhill from the base of the flagpole is approximately 55.4 feet.
altitude of top of flagpole
= 40+72tan17
length of cable:
length^2=altitude^2+72^2