A tree grows at an angle of 4^∘ from the vertical due to prevailing winds. At a point 40 meters

step 1 Working off the figure in the diagram, we get this triangle with sign of 30 degrees, the height.

A tree grows at an angle of 4^∘ from the vertical due to...

A tree grows at an angle of $4^{\circ}$ from the vertical due to prevailing winds. At a point 40 meters from the base of the tree, the angle of elevation to the top of the tree is $30^{\circ}$ (see figure). (a) Write an equation that you can use to find the height $h$ of the tree. (b) Find the height of the tree. CAN'T COPY THE FIGURE

A tree grows at an angle of $4^{\circ}$ from the vertical due to prevailing winds. At a point 40 meters from the base of the tree, the angle of elevation to the top of the tree is $30^{\circ}$ (see figure).
(a) Write an equation that you can use to find the height $h$ of the tree.
(b) Find the height of the tree.
CAN'T COPY THE FIGURE

Key Concepts

Angle of Elevation

This concept refers to the angle between the horizontal line from an observer and the line of sight to an object above the observer’s eye level. In problems like this, the angle of elevation helps relate the vertical height of the object to the horizontal distance from the observer, making it foundational for setting up trigonometric equations in right?triangle situations.

Trigonometric Ratios in Right Triangles

Trigonometric functions, especially the tangent function, are crucial in relating the angles of a right triangle to the ratios of its sides. In this context, the tangent of an angle is used to link the vertical component (height) and the horizontal distance, allowing the formation of an equation that can be solved for the unknown height.

Decomposition of Inclined Lines

When an object (like a leaning tree) is not vertical, its actual length differs from its vertical height. This discrepancy is resolved by decomposing the object’s length into vertical and horizontal components using cosine and sine of the lean angle. This idea is essential to adjust the measurements from the inclined object to the corresponding right triangle used in trigonometric calculations.

Equation Setup and Algebraic Manipulation

Once the right triangle is identified and the components are expressed via trigonometric functions, the next step is setting up an equation that relates the known measurements (such as the distance from the observer and the angle of elevation) to the unknown (the tree’s height). This typically involves writing an equation using the tangent of the given angle and then solving for the unknown height through algebraic manipulation.

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