A 127 foot tower is located on hill that is inclined 38° to the horizontal. A guy wire is to be attached to the top of the tower and anchored at a point 64 feet downhill from the base of the tower. Find the length of wire needed.
law of cosines
x² = 127² + 64² - [2 * 127 * 64 * cos(90º + 38º)]
draw the figure.
Label tower 127', then from the top of tower to the 64 mark downill, length L, draw a horzontal from the 64 mark, and then the height from the horizonal line to the FOOT of the tower as A.
Label the angles.
Note the angle formed by the hill and the base of the tower (the internal angle)can be determined from the vertical line through the base: Angleinternal = 90+38 Think that out.
So, you have two sides of on the guy wire triangle L, and 127, and 64.
You have SAS. use the law of cosines:
L^2=127^2+64^2-2*127*64*CosAngleinternal. solve for L
Well, first things first, let's appreciate the fact that this tower is on a hill inclined at 38°. That's hill-arious!
Now, on to the wire business. To find the length of the wire needed, we can create a right triangle with the tower as the vertical side, the distance downhill as the horizontal side, and the wire as the hypotenuse.
Using some trigonometry wizardry, we can use the sine function to find the length of the wire. The sine of an angle is equal to the ratio of the opposite side (tower height) to the hypotenuse (wire length).
So, let's crunch some numbers. The opposite side (tower height) is 127 feet, and the angle is 38°. Using the sine function, we can calculate:
sine(38°) = tower height / wire length
Wire length = tower height / sine(38°)
Wire length = 127 feet / sine(38°)
Using a calculator (or my superior clown math skills), I calculate the wire length to be approximately 205.47 feet.
So, the wire length needed is around 205.47 feet. Just make sure to fasten it securely, we don't want any clowning around with safety!
To find the length of the guy wire needed, we can use the concept of right triangles.
First, let's draw a diagram to visualize the situation:
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In this diagram, the tower is represented by a vertical line. The guy wire is represented by a slanted line (the hypotenuse of a right triangle), and the downhill distance is represented by a horizontal line.
We are given:
- The height of the tower is 127 feet.
- The angle of inclination of the hill is 38 degrees.
- The distance downhill from the base of the tower to the anchor point is 64 feet.
Let's label the relevant lengths in the diagram:
- The vertical length (height of the tower) is 127 feet.
- The horizontal length (downhill distance) is 64 feet.
- The slanted length (length of the guy wire) is what we need to find.
Now, using trigonometry, we can use the sine function to relate the angle and the lengths of the triangle:
sin(angle) = opposite/hypotenuse
Applying this to our situation:
sin(38°) = 127/hypotenuse
To find the length of the guy wire (hypotenuse), we rearrange this equation:
hypotenuse = 127/sin(38°)
Using a calculator to evaluate the sine of 38 degrees:
sin(38°) ≈ 0.61566
Now we can find the length of the guy wire:
hypotenuse ≈ 127/0.61566
hypotenuse ≈ 206.41
Therefore, the length of wire needed is approximately 206.41 feet.
To find the length of the guy wire needed, we can use the concept of right triangles and trigonometry.
First, let's visualize the problem. We have a tower located on a hill that is inclined 38° to the horizontal. The tower is 127 feet tall. We need to attach a guy wire from the top of the tower to a point 64 feet downhill from the base of the tower.
Let's draw a diagram to understand the situation:
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64 ft
|<---distance --->|
In the above diagram, the vertical line represents the height of the tower (127 ft), the horizontal line represents the distance from the base of the tower to the anchor point (64 ft), and the diagonal line represents the guy wire we are trying to find.
We can see that we have a right triangle formed by the tower, the distance from the tower to the anchor point, and the guy wire. To find the length of the guy wire, we can use the trigonometric function cosine (cos). The cosine of an angle is defined as the adjacent side divided by the hypotenuse.
In this case, the adjacent side is the distance from the base of the tower to the anchor point (64 ft), and the hypotenuse is the length of the guy wire (which we need to find).
Using the cosine formula:
cos(38°) = adjacent / hypotenuse
We can rearrange the formula to solve for the hypotenuse (length of the guy wire):
hypotenuse = adjacent / cos(38°)
Plugging in the values:
hypotenuse = 64 ft / cos(38°)
Using a calculator, we can find the cosine of 38°, which is approximately 0.7880.
hypotenuse = 64 ft / 0.7880
hypotenuse ≈ 81.22 ft
Therefore, the length of the guy wire needed is approximately 81.22 feet.