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A 2020-foot fall flagpole is leaning towards a house. If the flagpole makes an 85∘85^\circ angle with the ground and the angle of elevation from the base of the house to the top of the pole is 5555 degrees, find the distance from the base of the flagpole to the house.

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Q. A 2020-foot fall flagpole is leaning towards a house. If the flagpole makes an 85∘85^\circ angle with the ground and the angle of elevation from the base of the house to the top of the pole is 5555 degrees, find the distance from the base of the flagpole to the house.
  1. Identify Angles and Sides: Identify the angles and sides of the right triangle formed by the flagpole, the ground, and the line from the base of the flagpole to the house.\newlineWe know:\newline- The flagpole makes an 8585-degree angle with the ground.\newline- The angle of elevation from the base of the house to the top of the pole is 5555 degrees.\newline- The length of the flagpole is 2020 feet.
  2. Determine Angle Between: Determine the angle between the flagpole and the line from the base of the house to the top of the pole.\newlineSince the flagpole makes an 8585-degree angle with the ground, and the angle of elevation is 5555 degrees, the angle between the flagpole and the line from the base of the house to the top of the pole is 180−85−55180 - 85 - 55 degrees.\newlineCalculation: 180−85−55=40180 - 85 - 55 = 40 degrees.
  3. Use Law of Sines: Use the law of sines to find the distance from the base of the flagpole to the house.\newlineThe law of sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all sides and angles in the triangle.\newlineLet xx be the distance from the base of the flagpole to the house.\newlineWe have:\newlinesin⁡(85)20=sin⁡(40)x\frac{\sin(85)}{20} = \frac{\sin(40)}{x}
  4. Solve for Distance: Solve for xx, the distance from the base of the flagpole to the house.\newlineCross-multiply to get:\newline20×sin⁡(40)=x×sin⁡(85)20 \times \sin(40) = x \times \sin(85)\newlineNow, divide both sides by sin⁡(85)\sin(85) to isolate xx:\newlinex=20×sin⁡(40)sin⁡(85)x = \frac{20 \times \sin(40)}{\sin(85)}
  5. Calculate Value of x: Calculate the value of x using a calculator.\newlineCalculation:\newlinex≈(20×0.6428)/0.9962x \approx (20 \times 0.6428) / 0.9962\newlinex≈12.856/0.9962x \approx 12.856 / 0.9962\newlinex≈12.91x \approx 12.91 feet

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