Problem 49 - Algebra and Trigonometry

Question

A 20-ft ladder leans against a building so that the angle between the ground and the ladder is \(72^{\circ}.\) How high does the ladder reach on the building?

Step-by-Step Solution

Answer

The ladder reaches approximately 19.022 feet high on the building.

Step 1: Understanding the Scenario

We need to find out how high the ladder reaches on the building. The ladder forms a right triangle with the ground and the wall of the building where the ladder is the hypotenuse.

Step 2: Identify Known Values

We know that:- The length of the ladder (hypotenuse) is 20 ft.- The angle between the ground and the ladder is \(72^{\circ}\).

Step 3: Establishing the Right Trigonometric Function

To find the height reached by the ladder on the building, we can use the sine function. The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse.

Step 4: Apply the Sine Function Formula

Using the sine function for angle \(72^{\circ}\), we have:\[\sin(72^{\circ}) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{20}\]where \(h\) is the height we want to find.

Step 5: Solve for the Height

Rearrange the equation to solve for \(h\):\[ h = 20 \times \sin(72^{\circ}) \]Using a calculator, \(\sin(72^{\circ}) \approx 0.9511\). So,\[ h = 20 \times 0.9511 \approx 19.022 \text{ ft} \]

Step 6: Conclusion

The height which the ladder reaches on the building is approximately \(19.022\) feet.

Key Concepts

Angle Calculation

When solving problems involving right triangles, calculating angles is a common challenge. In this scenario, we have an angle of \(72^{\circ}\). This represents the angle between the ladder and the ground. Understanding angles is key because they inform us how steep or shallow the ladder is positioned against the building.

• Geometric angles are measured in degrees or radians.
• In a right triangle, the sum of all angles equals \(180^{\circ}\).
• One of these angles is always \(90^{\circ}\) by definition, known as the right angle.

For angle calculation, we often need other mathematical tools like trigonometric functions when dealing with right triangles. These functions rely heavily on angle measurement.

Right Triangle

A right triangle is a type of triangle that includes one \(90^{\circ}\) angle. This feature is crucial because it allows us to use specific trigonometric ratios to solve for unknown values within the triangle. These triangles have three sides:

• The hypotenuse: This is the longest side, opposite the right angle.
• The opposite side: This is the side opposite the angle of interest (not the right angle).
• The adjacent side: This is the side next to the angle of interest (again, not the right angle).

The presence of a right angle provides the foundational reason why trigonometric functions like sine, cosine, and tangent are applicable. In our ladder problem, the ladder itself forms the hypotenuse of the right triangle.

Sine Function

The sine function is one of the primary trigonometric functions used to relate the angles and sides of a right triangle. Its formula is expressed as:\[\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\]Where:

• \(\sin(\theta)\) represents the sine of angle \(\theta\).
• The opposite side is the side opposite the angle \(\theta\).
• The hypotenuse is the triangle's longest side.

In the context of our ladder problem, we use the sine function to calculate how high up the building the ladder extends. For angle \(72^{\circ}\), the sine gives us the ratio needed to find the ladder's reach on the building using the formula \(h = 20 \times \sin(72^{\circ})\).

Hypotenuse

In any right triangle, the hypotenuse is the longest side and is instrumental when using trigonometric ratios. It is directly opposite the triangle's right angle. Leveraging the hypotenuse is crucial for calculations involving the Pythagorean theorem or trigonometric functions.

• In our exercise, the hypotenuse is the 20-ft ladder.
• This side is constant regardless of the angle changes, unless the setup of the physical scenario is altered.
• All trigonometric functions in a right triangle are based on their relationship to the hypotenuse.

Understanding the role of the hypotenuse not only aids in solving triangle problems but also in grasping broader geometric principles.