Problem 41 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A flag is mounted on the semicircular dome of radius \(\mathrm{r}\). The elevation of the top of the flag at any point on the ground is \(30^{\circ}\). Moving d distance towards the dome, when the flag is just visible, the angle of elevation is \(45^{\circ}\). The relation between \(\mathrm{r}\) and \(\mathrm{d}\) is (A) \(r=\frac{d}{\sqrt{2}(\sqrt{3}-1)}\) (B) \(\mathrm{r}=d \frac{2 \sqrt{2}}{\sqrt{3}+1}\) (C) \(\mathrm{r}=\frac{d}{\sqrt{2}(\sqrt{3}+1)}\) (D) \(\mathrm{r}=d \frac{2 \sqrt{2}}{\sqrt{3}-1}\)

Step-by-Step Solution

Answer

Option C: \( r = \frac{d}{\sqrt{2}(\sqrt{3}+1)} \).

Step 1: Sketch the Problem

Start by drawing a semicircle to represent the dome with radius \( r \), and place a flagpole at the peak. Then mark two points on the ground, one where the angle of elevation to the top of the flag is \( 30^{\circ} \) and another, closer to the dome, where the angle is \( 45^{\circ} \). Denote the second point as being a distance \( d \) closer to the dome than the first point.

Step 2: Introduce the Variables

Assign coordinates for the positions: let the center of the dome be the origin \((0, 0)\), and the height of the flag be \( h \). The ground point where the angle is \( 30^{\circ} \) is \((x_1, 0)\) and where the angle is \( 45^{\circ} \) is \((x_1-d, 0)\).

Step 3: Apply the Tangent Function to Both Scenarios

For the \(30^{\circ}\) angle, \( \tan(30^{\circ}) = \frac{h}{x_1} \). For the \(45^{\circ}\) angle, \( \tan(45^{\circ}) = \frac{h}{x_1-d} \). Use these relationships to set up two equations.

Step 4: Solve for \( h \) in Each Equation

From \( \tan(30^{\circ}) = \frac{h}{x_1} \), get \( h = x_1 \cdot \frac{1}{\sqrt{3}} \). From \( \tan(45^{\circ}) = \frac{h}{x_1-d} \), solve it as \( h = x_1-d \). Now, since both expressions equal \( h \), set them equal to each other.

Step 5: Equate and Solve Equations

Set the two expressions for \( h \) equal: \( x_1 \cdot \frac{1}{\sqrt{3}} = x_1 - d \). Solve this equation for \( x_1 \), finding \( x_1 = \frac{d}{1 - \frac{1}{\sqrt{3}}} \), simplifying to \( x_1 = \frac{d \cdot \sqrt{3}}{\sqrt{3} - 1} \).

Step 6: Express \( r \) In Terms of \( d \)

From the geometry of the problem, know that \( r = x_1 - d \). Substitute for \( x_1 \) from the previous step, yielding \( r = \frac{d \cdot \sqrt{3}}{\sqrt{3} - 1} - d \). Simplify this, using algebraic techniques to find \( r = \frac{d}{\sqrt{2}(\sqrt{3} +1)} \).

Step 7: Identify the Correct Option

Match this expression for \( r \) with one of the given options. It corresponds to \( \text{Option C}: r = \frac{d}{\sqrt{2}(\sqrt{3}+1)} \).

Key Concepts

Angle of Elevation

The angle of elevation is the angle between the horizontal ground and the line of sight to an object above the ground level. This concept is frequently used in trigonometry to determine heights and distances that are not easy to measure directly.
When dealing with angles of elevation:

• The observer's position is always horizontal or level with the base of the observed object.
• The line of sight rises from the horizontal to the target object, forming the angle of elevation.
• Typical angles of elevation are often discussed or calculated using right triangles and trigonometric functions.

For this problem, we first observe the flag from a distance where the angle of elevation is 30 degrees, and then move closer until the angle increases to 45 degrees. These angles describe two different lines of sight: one longer and flatter, and another shorter and steeper. By applying trigonometric principles, you can utilize these angles of elevation to find unknown distances in this context.

Tangent Function

The tangent function, represented by \( \tan \theta \), is one of the primary trigonometric functions that relate the angles of a right triangle to its side lengths. Specifically, the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. For example, in a right triangle, if \( \theta \) is one of the angles, then \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
For the exercise, the tangent function is applied to gauge the height of the flag relative to specific points on the ground:

• The formula \( \tan(30^{\circ}) = \frac{h}{x_1} \) is used for the first observation.
• For the second observation, \( \tan(45^{\circ}) = \frac{h}{x_1-d} \) is applied.
• These set equations, once solved, can help find the unknown height \( h \) and location on the horizontal plane.

Using these equations helps unravel how changes in an observer’s position (closer or further from an object) affect the angle, thereby determining the exact dimensions related to the problem.

Semicircle Geometry

Understanding semicircle geometry is crucial in solving problems related to objects situated on curved structures like domes. A semicircle is half of a circle, defined by its diameter stretching across its widest span. Here are some key points:

• A semicircle's radius is simply the radius of the original circle of which the semicircle is a portion.
• On a real-world dome, this would mean extending from the center point of the base to any point along the curved surface.
• Geometric problems often involve calculating paths or projections involving distances or angles based on these radii.

The context of the problem involves a flag on top of a semicircular dome. The flag's height is linked to the dome's radius. By using the semicircle's geometry combined with the points on the ground (established in the problem), one can use trigonometric calculations to find relationships like the distance from any given point to the base of the dome, which directly influences solving for \( r \), the radius. This geometric comprehension, coupled with trigonometry, offers invaluable insights into resolving complex spatial problems.