Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

A gondola tower in an amusement park casts a shadow that is 80 feet long, while a sign that is 4 feet tall casts a shadow that is \(3 \frac{1}{2}\) feet long. Draw a diagram for the situation. Then find the height of the tower.

Step-by-Step Solution

Answer

The height of the gondola tower is approximately 91.43 feet.

Step 1: Convert Measurements to the Same Unit

Convert the shadow length of the sign which is \(3 \frac{1}{2}\) feet into the decimal system for easier calculation. \(3 \frac{1}{2}\) feet is equal to 3.5 feet.

Step 2: Set Up the Proportional Relationship

Set up the proportion using the principle of similar triangles, that is, the ratio of the corresponding sides of two similar triangles is equal:\(\frac{height of tower}{length of tower's shadow} = \frac{height of sign}{length of sign's shadow}\)Substitute the given values:\(\frac{height of tower}{80ft} = \frac{4ft}{3.5ft}\)

Step 3: Solve for the Unknown

Solve the proportion for the height of the tower by cross-multiplication and division:\(\frac{height of tower}{80ft} = \frac{4ft}{3.5ft}\) Cross multiply:\(Height of tower * 3.5ft = 80ft * 4ft\)Divide both sides by 3.5ft to get:\(Height of tower = \frac{(80ft * 4ft)}{3.5ft}\)Calculate the height to get the final answer.

Key Concepts

Understanding Shade and Shadow

Shade and shadow play a crucial role in understanding the problem of finding the height of the gondola tower. When the sun shines, objects cast shadows on the ground. The length of these shadows depends on the angle of the sunlight and the height of the objects. Simply put, as the object or the light source moves, the shadow changes.
This concept helps us analyze the problem of the gondola tower and the sign because, in a given timeframe with a fixed light source, similar objects will cast proportional shadows, allowing us to make calculated guesses about their heights based on shadow lengths.
In our case, the shadows of the gondola tower and the sign provide a basis for comparison, leading to solving the problem through similar triangles.

Proportional Reasoning in Triangles

To solve problems involving shadows, such as finding the unknown height of a tower, we use the concept of proportional reasoning. Proportional reasoning is about understanding and using ratios to find unknown measurements.
When two objects—like a tower and a sign—create shadows, they form triangles with the ground. These triangles can be considered similar if they have the same shape, regardless of size.

• The concept of similar triangles is built upon the principle that corresponding angles in these triangles are equal.
• The sides of similar triangles are proportional, meaning the ratio of one pair of corresponding sides is the same as the ratio of another pair.

If the triangles are similar, we can set up a proportion between their sides. For the gondola tower and sign, the proportion would be between their heights and their respective shadow lengths. This line of thought leads us to set up equations, helping find the unknown height of the tower.

Cross Multiplication Technique

Cross multiplication is a straightforward technique to solve proportions—equations that state two ratios are equal. In the context of our problem, cross multiplication helps us find the height of the gondola tower.
After establishing a proportion between the height and shadow of the sign and the height and shadow of the tower, we cross-multiply to isolate the unknown.

• Start by setting the equation as a/b = c/d, where a, b, c, and d represent the known and unknown values.
• Then cross-multiply, giving us ad = bc.
• This method simplifies the equation, making it easier to solve for the unknown value.

In our case, we set up the proportion based on similar triangles, cross-multiply, and solve for the tower's height. This directs us logically through the equations until we efficiently reach the answer.