Draw a diagram and solve please

Question

Joe wants to put Christmas lights on the edge of his roof. His roof makes a 45 degree angle with the top of the house. If the house is 120 feet wide, find the amount of roof that needs to be covered by lights. Draw a diagram and solve the problem.

Answer 1

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/45° |

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120 ft

1. We need to find the length of one side of the roof.
2. Let's denote the length of one side of the roof as L.
3. Each half of the roof forms a right triangle with the width divided into two equal parts of 60 feet (since 120 ft/2=60 ft).
4. The angle between the roof and the horizontal line is 45 degrees.
5. Trigonometric Relation:
6. Using the tangent function: tan⁡(45∘)=opposite / adjacent=height / 60 ft

Since Tan 45 = 1

1 = height / 60 ft

height = 60 ft

1. Find the Length of the Hypotenuse:
2. Using the Pythagorean theorem for the right triangle,

L=sqrt((height)^2+(half width)^2)

L=sqrt((60 ft)^2+(60 ft)^2)

L=sqrt(3600 ft+3600 ft)

L = sqrt(7200)

L = 60 sqrt(2)

Final Answer:

The total length of the roof that needs to be covered by Christmas lights is 120 sqrt(2)

Converting sqrt{2}​ to a decimal (approximately 1.414),

120×1.414≈169.68 ft

So, Joe needs to cover approximately 169.68 feet of roof with Christmas lights.