A boat sails directly toward a 150-meter tall skyscraper that stands on the edge of a harbor. Let xx be the horizontal distance between the base of the building and the boat. The angle θ, measured in radians, is the angle of elevation from the boat to the top of the building.

If the angle of elevation is π4\frac{\pi}{4} or 45∘45^{\circ}, how far from the building is the boat? (Note: It might be helpful to sketch a picture of the situation to determine an equation for this situation.)

The boat is [Fill in the blank],
meters from the building when the angle of elevation is π4\frac{\pi}{4} or 45∘45^{\circ}.

Question

A boat sails directly toward a 150-meter tall skyscraper that stands on the edge of a harbor. Let xx be the horizontal distance between the base of the building and the boat. The angle θ, measured in radians, is the angle of elevation from the boat to the top of the building.

If the angle of elevation is π4\frac{\pi}{4} or 45∘45^{\circ}, how far from the building is the boat? (Note: It might be helpful to sketch a picture of the situation to determine an equation for this situation.)

The boat is [Fill in the blank],
 meters from the building when the angle of elevation is π4\frac{\pi}{4} or 45∘45^{\circ}.

BlackTom AI · Accepted Answer

The problem describes a right triangle formed by the boat, the base of the skyscraper, and the top of the skyscraper. The height (opposite side) is 150 meters, and the horizontal distance from the boat to the building's base is x (the adjacent side). The angle of elevation θ is given as π/4 (which is 45 degrees), so we use the tangent ratio: tan(θ) = opposite / adjacent = 150 / x.

Since tan(π/4) = 1, we have 1 = 150 / x. Solving for x gives x = 150.

This means the horizontal distance from the boat to the base of the building is 150 meters. If we break down the reasoning into steps:
- Identify the right triangle and label sides: opposite = 150 m, adjacent = x, angle θ = π/4.
- Apply tan(π/4) = opposite/adjacent.
- Substitute and solve: 1 = 150/x → x = 150.
- Conclude that the distance from the boat to the building's base is 150 meters.

Note: Because there were no provided multiple-choice options, the explanation focuses on the geometric relation and the standard trigonometric identity for a 45-degree angle.

DIFFERENTIAL CALCULUS (MTH_251_020_W2025) Unit 4 - Inverse Functions - Preclass

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