Ladder Height Problem Solving How Far Up A Building Does A 12 Foot Ladder Reach
In the realm of mathematics, particularly trigonometry and geometry, practical problems often require us to apply theoretical concepts to real-world scenarios. One such common scenario involves ladders leaning against buildings, forming right triangles. Solving these problems involves using trigonometric ratios or the Pythagorean theorem to find unknown lengths or angles. In this comprehensive exploration, we will delve into a specific problem: determining how far up a building a 12-foot ladder reaches when it makes a 45-degree angle with the building. This problem is not just a mathematical exercise; it's a fundamental concept applicable in various fields, including architecture, construction, and engineering. Understanding the principles behind this problem can enhance your problem-solving skills and provide valuable insights into the practical applications of trigonometry.
Our focus is on the classic ladder-against-a-building scenario. Imagine a 12-foot ladder leaning against the wall of a building. The ladder forms an angle of 45 degrees with the building itself. The question we aim to answer is: How far up the building does the ladder reach? This is a typical problem that tests our understanding of trigonometric ratios and their application in right-angled triangles. The ladder, the building, and the ground form a right triangle, where the ladder is the hypotenuse, the building's wall is one leg, and the ground is the other leg. The angle between the ladder and the building is given, and we need to find the length of the leg representing the height the ladder reaches on the building.
Before we dive into solving the problem, it's crucial to understand the basic trigonometric ratios: sine, cosine, and tangent. These ratios relate the angles of a right triangle to the lengths of its sides. In a right triangle:
- Sine (sin) of an angle is the ratio of the length of the opposite side to the hypotenuse.
- Cosine (cos) of an angle is the ratio of the length of the adjacent side to the hypotenuse.
- Tangent (tan) of an angle is the ratio of the length of the opposite side to the adjacent side.
In our problem, we have the angle between the ladder and the building (45 degrees) and the length of the ladder (the hypotenuse). We need to find the height the ladder reaches on the building, which is the side adjacent to the given angle. Therefore, the cosine function is the most appropriate trigonometric ratio to use in this case. The cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse, which perfectly fits our known and unknown variables. By understanding these ratios, we can effectively translate real-world problems into mathematical equations and solve them.
Now that we understand the trigonometric ratios, let's apply this knowledge to our problem. We have a right triangle formed by the ladder, the building, and the ground. The ladder, which is 12 feet long, acts as the hypotenuse of this triangle. The angle between the ladder and the building is 45 degrees. We are trying to find the height the ladder reaches on the building, which we will denote as 'h'. This height is the side adjacent to the 45-degree angle. As we discussed earlier, the cosine function relates the adjacent side to the hypotenuse. Therefore, we can set up the following trigonometric equation:
This equation forms the foundation for solving our problem. It translates the geometric relationships into a mathematical expression that we can manipulate to find the unknown height 'h'. The next step is to evaluate the cosine of 45 degrees and then solve for 'h'. This process demonstrates the power of trigonometry in bridging the gap between angles and lengths in geometric figures.
The next crucial step in solving our problem is to determine the value of the cosine of 45 degrees. The cosine of 45 degrees is a well-known trigonometric value, and it is equal to $\frac{\sqrt{2}}{2}$. This value is derived from the special right triangle known as the 45-45-90 triangle, which has angles of 45 degrees, 45 degrees, and 90 degrees. In this triangle, the two legs are of equal length, and the hypotenuse is $\sqrt{2}$ times the length of each leg. The trigonometric values for these special angles (30, 45, and 60 degrees) are fundamental in trigonometry and appear frequently in various mathematical and real-world problems. It's beneficial to memorize these values or understand how to derive them using geometric properties. Now that we know $\cos(45^\circ) = \frac{\sqrt{2}}{2}$, we can substitute this value into our equation and proceed to solve for the height.
Having established the trigonometric equation and evaluated the cosine of 45 degrees, we can now solve for the height 'h'. Our equation is:
To isolate 'h' and find its value, we need to multiply both sides of the equation by 12. This step is based on the fundamental algebraic principle that allows us to perform the same operation on both sides of an equation without changing its balance. By multiplying both sides by 12, we effectively eliminate the denominator on the right side of the equation, leaving 'h' by itself. The equation now becomes:
This step simplifies the equation significantly and brings us closer to the final solution. The next step involves simplifying the expression on the right-hand side to find the numerical value of 'h'. This process illustrates how algebraic manipulation combined with trigonometric knowledge allows us to solve real-world problems efficiently.
Continuing from the equation $h = 12 \times \frac{\sqrt{2}}{2}$, we need to simplify the expression on the right-hand side to find the value of 'h'. We can start by dividing 12 by 2, which gives us 6. The equation then becomes:
This simplified form gives us the height 'h' in terms of a radical. The value $6\sqrt{2}$ represents the exact height the ladder reaches on the building. To get a sense of the numerical value, we can approximate the square root of 2, which is approximately 1.414. Multiplying 6 by 1.414 gives us an approximate value for 'h'. However, for the purpose of selecting the correct answer from the given options, the exact form $6\sqrt{2}$ is sufficient. This simplification process highlights the importance of algebraic skills in handling mathematical expressions and arriving at a clear and concise solution.
After simplifying the equation, we have found that the height 'h' the ladder reaches on the building is $6\sqrt{2}$ feet. Looking at the given options:
A. $12 \sqrt{2}$ feet B. $6 \sqrt{2}$ feet C. 6 feet D. $24 \sqrt{2}$ feet
We can see that option B, $6 \sqrt{2}$ feet, matches our calculated height. Therefore, the correct answer is B. $6 \sqrt{2}$ feet. This final step confirms our solution and demonstrates the logical progression from the initial problem statement to the final answer. The process involved understanding trigonometric ratios, setting up a trigonometric equation, evaluating the cosine of an angle, and simplifying the expression to find the unknown height. This problem-solving approach is applicable to various scenarios involving right triangles and trigonometric functions.
In conclusion, we have successfully determined the height a 12-foot ladder reaches on a building when it forms a 45-degree angle with the building. By applying trigonometric principles, specifically the cosine function, we were able to set up an equation and solve for the unknown height. The correct answer, $6 \sqrt{2}$ feet, demonstrates the practical application of trigonometry in real-world scenarios. This problem not only reinforces our understanding of trigonometric ratios but also highlights the importance of algebraic manipulation and problem-solving skills in mathematics. Understanding these concepts is crucial for various fields, including engineering, architecture, and construction, where accurate calculations are essential. Mastering such problems builds a strong foundation in mathematics and enhances our ability to tackle more complex challenges.
