Bill determines that the angle of elevation to the top of a building measures 40°. If he walks 100 m closer to the building, the measure of the new angle of elevation will be 50°.

Answer 1

Let's assume that the distance from Bill to the building is x meters.
According to the given information, the angle of elevation to the top of the building measures 40°.
Using trigonometry, we can determine that the height of the building is x * tan(40°).

When Bill walks 100 m closer to the building, the new distance from him to the building becomes (x - 100) meters.
According to the new information, the measure of the new angle of elevation will be 50°.
Using trigonometry again, we can determine that the new height of the building is (x - 100) * tan(50°).

Setting the two heights equal to each other, we have x * tan(40°) = (x - 100) * tan(50°).

Simplifying the equation, we get x * tan(40°) = x * tan(50°) - 100 * tan(50°).

Adding 100 * tan(50°) to both sides, we have x * tan(40°) + 100 * tan(50°) = x * tan(50°).

Dividing both sides by tan(40°), we get x + 100 * tan(50°) / tan(40°) = x * tan(50°) / tan(40°).

Subtracting x * tan(50°) / tan(40°) from both sides, we have x + 100 * tan(50°) / tan(40°) - x * tan(50°) / tan(40°) = 0.

Combining like terms, we get x * (1 - tan(50°) / tan(40°)) + 100 * tan(50°) / tan(40°) = 0.

Multiplying both sides by tan(40°), we have x * (tan(40°) - tan(50°)) + 100 * tan(50°) = 0.

Now, we will solve for x using the values of the tangent function from a calculator to approximate the values of tan(40°) and tan(50°):

x * (0.8391 - 1.1918) + 100 * 1.1918 = 0,
x * (-0.3527) + 119.18 = 0,
-0.3527x + 119.18 = 0,
-0.3527x = -119.18,
x = -119.18 / -0.3527,
x ≈ 337.98.

Therefore, the distance from Bill to the building is approximately 337.98 meters.

Answer 2

no question was asked, but you did find the initial distance to the building.

To find the height h of the building, solve
h cot40° - h cot50° = 100
h = 283.564

Answer 3

Apologies for the confusion. Based on the given information and using trigonometry, the height of the building is approximately 283.564 meters.