Math / Geometry

QuestionThe figure shows a 350 foot tower on the side of a hill that forms a $5^{\circ}$ angle with the horizontal. Find the length of each of the two guy wires that are anchored 110 feet uphill and downhill from the tower's base and extend to the top of the tower.
Part (a) What is the length of the uphill guy wire? 357.6 feet (Round to the nearest tenth as needed.)
Part (b) What is the length of the downhill guy wire? $\square$ feet (Round to the nearest tenth as needed.)

Studdy Solution

STEP 1

What is this asking? We need to find the lengths of two wires supporting a tower on a hill, one anchored uphill and the other downhill. Watch out! Don't forget to account for the hill's angle when calculating the wire lengths.
It's not just a straight up-and-down triangle!

STEP 2

1. Visualize and label
2. Uphill wire length
3. Downhill wire length

STEP 3

Imagine the tower, hill, and uphill wire forming a triangle.
Label the tower height as $350$ feet.
The horizontal distance from the tower base to the uphill anchor is $110$ feet.
The angle between the hill and the horizontal is $5^\circ$ .

STEP 4

We can find the distance along the hill to the anchor point using trigonometry.
Since $\tan(5^\circ) = \frac{\text{vertical distance}}{\text{horizontal distance}}$ , the vertical distance is $110 \cdot \tan(5^\circ) \approx 9.60$ feet.
This means the hill length to the uphill anchor is $\sqrt{110^2 + 9.60^2} \approx 110.35$ feet using the Pythagorean theorem.

STEP 5

Now imagine the downhill scenario.
The tower height remains $350$ feet, the horizontal distance is still $110$ feet, but the hill slopes down.
The vertical distance is the same as calculated before, $9.60$ feet, but this time it's downwards.
The hill length to the downhill anchor is also the same, approximately $110.35$ feet.

STEP 6

The top of the tower is $350$ feet vertically above its base, and the uphill anchor is $9.60$ feet vertically below the base.
So, the vertical distance between the top of the tower and the uphill anchor is $350 + 9.60 = 359.60$ feet.

STEP 7

Now we have a right triangle with legs of $110$ feet (horizontal distance) and $359.60$ feet (vertical distance).
The uphill wire length is the hypotenuse.
Using the Pythagorean theorem, the uphill wire length is $\sqrt{110^2 + 359.60^2} \approx 376.0$ feet.

STEP 8

The top of the tower is $350$ feet vertically above its base, and the downhill anchor is $9.60$ feet vertically below the base.
So, the vertical distance between the top of the tower and the downhill anchor is $350 - 9.60 = 340.40$ feet.

STEP 9

We have another right triangle with legs of $110$ feet (horizontal distance) and $340.40$ feet (vertical distance).
The downhill wire is the hypotenuse.
Using the Pythagorean theorem, the downhill wire length is $\sqrt{110^2 + 340.40^2} \approx 357.6$ feet.

STEP 10

(a) The uphill guy wire is approximately 376.0 feet long. (b) The downhill guy wire is approximately 357.6 feet long.

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Studdy Solution

STEP 1

What is this asking? We need to find the lengths of two wires supporting a tower on a hill, one anchored uphill and the other downhill. Watch out! Don't forget to account for the hill's angle when calculating the wire lengths.
It's not just a straight up-and-down triangle!

STEP 2

1. Visualize and label
2. Uphill wire length
3. Downhill wire length

STEP 3

Imagine the tower, hill, and uphill wire forming a triangle.
Label the tower height as $350$ feet.
The horizontal distance from the tower base to the uphill anchor is $110$ feet.
The angle between the hill and the horizontal is $5^\circ$ .

STEP 4

STEP 5

STEP 6

The top of the tower is $350$ feet vertically above its base, and the uphill anchor is $9.60$ feet vertically below the base.
So, the vertical distance between the top of the tower and the uphill anchor is $350 + 9.60 = 359.60$ feet.

STEP 7

Now we have a right triangle with legs of $110$ feet (horizontal distance) and $359.60$ feet (vertical distance).
The uphill wire length is the hypotenuse.
Using the Pythagorean theorem, the uphill wire length is $\sqrt{110^2 + 359.60^2} \approx 376.0$ feet.

STEP 8

The top of the tower is $350$ feet vertically above its base, and the downhill anchor is $9.60$ feet vertically below the base.
So, the vertical distance between the top of the tower and the downhill anchor is $350 - 9.60 = 340.40$ feet.

STEP 9

We have another right triangle with legs of $110$ feet (horizontal distance) and $340.40$ feet (vertical distance).
The downhill wire is the hypotenuse.
Using the Pythagorean theorem, the downhill wire length is $\sqrt{110^2 + 340.40^2} \approx 357.6$ feet.

STEP 10

(a) The uphill guy wire is approximately 376.0 feet long. (b) The downhill guy wire is approximately 357.6 feet long.

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Studdy Solution

STEP 1

What is this asking? We need to find the lengths of two wires supporting a tower on a hill, one anchored uphill and the other downhill. Watch out! Don't forget to account for the hill's angle when calculating the wire lengths.It's not just a straight up-and-down triangle!

STEP 2

1. Visualize and label2. Uphill wire length3. Downhill wire length

STEP 3

Imagine the tower, hill, and uphill wire forming a triangle.Label the tower height as 350350350 feet.The horizontal distance from the tower base to the uphill anchor is 110110110 feet.The angle between the hill and the horizontal is 5∘5^\circ5∘.

STEP 4

STEP 5

STEP 6

The top of the tower is 350350350 feet vertically above its base, and the uphill anchor is 9.609.609.60 feet vertically below the base.So, the vertical distance between the top of the tower and the uphill anchor is 350+9.60=359.60350 + 9.60 = 359.60350+9.60=359.60 feet.

STEP 7

STEP 8

The top of the tower is 350350350 feet vertically above its base, and the downhill anchor is 9.609.609.60 feet vertically below the base.So, the vertical distance between the top of the tower and the downhill anchor is 350−9.60=340.40350 - 9.60 = 340.40350−9.60=340.40 feet.

STEP 9

STEP 10

**(a)** The uphill guy wire is approximately **376.0** feet long. **(b)** The downhill guy wire is approximately **357.6** feet long.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to find the lengths of two wires supporting a tower on a hill, one anchored uphill and the other downhill. Watch out! Don't forget to account for the hill's angle when calculating the wire lengths.
It's not just a straight up-and-down triangle!

1. Visualize and label
2. Uphill wire length
3. Downhill wire length

Imagine the tower, hill, and uphill wire forming a triangle.
Label the tower height as $350$ feet.
The horizontal distance from the tower base to the uphill anchor is $110$ feet.
The angle between the hill and the horizontal is $5^\circ$ .

The top of the tower is $350$ feet vertically above its base, and the uphill anchor is $9.60$ feet vertically below the base.
So, the vertical distance between the top of the tower and the uphill anchor is $350 + 9.60 = 359.60$ feet.

The top of the tower is $350$ feet vertically above its base, and the downhill anchor is $9.60$ feet vertically below the base.
So, the vertical distance between the top of the tower and the downhill anchor is $350 - 9.60 = 340.40$ feet.

(a) The uphill guy wire is approximately 376.0 feet long. (b) The downhill guy wire is approximately 357.6 feet long.