Problem 38 - Precalculus Student Solutions Manual 5th

Question

Problems involving the Pythagorean theorem have appeared in mathematics for thousands of years. This one is taken from the ancient Chinese work, Arithmetic in Nine Sections: There is a bamboo IO ft high, the upper end of which, being broken, reaches the ground 3 ft from the stem. Find the height of the break.

Step-by-Step Solution

Answer

The height of the break is about 4.55 feet.

Step 1: Understand the Problem

A bamboo that is initially 10 feet high is broken, and its top now touches the ground 3 feet away from the base. Your task is to find the height at which the bamboo is broken.

Step 2: Identify the Right Triangle

Visualize the problem as creating a right triangle where the vertical part of the bamboo is one leg, the horizontal distance (3 feet) is the other leg, and the slanted broken part is the hypotenuse.

Step 3: Set up the Pythagorean Theorem

Let the height of the break from the ground be denoted as 'h'. Since the bamboo was initially 10 feet tall, the hypotenuse (the length of the bamboo after breaking) will be \( 10 - h \) feet. Use the Pythagorean theorem, which states: \[ a^2 + b^2 = c^2 \]

Step 4: Substitute Known Values

In the Pythagorean theorem, let: \ a = h, \ b = 3 feet, \ c = 10 - h feet. Substitute these into the equation: \[ h^2 + 3^2 = (10 - h)^2 \]

Step 5: Simplify the Equation

Expand and simplify the equation: \[ h^2 + 9 = (10 - h)(10 - h) \] \[ h^2 + 9 = 100 - 20h + h^2 \]

Step 6: Isolate the Variable

Cancel out \ h^2 \ from both sides and solve for h: \[ 9 = 100 - 20h \] \[ 9 - 100 = -20h \] \[ -91 = -20h \] \[ h = \frac{91}{20} \]

Step 7: Calculate the Final Answer

Simplify the result to find the height of the break: \[ h = 4.55 \text{ feet} \]

Key Concepts

Right Triangle

To understand the problem, imagine the situation: a bamboo that was initially 10 feet high is broken, and its top touches the ground 3 feet away from the base. This creates a shape known in geometry as a right triangle.

• In a right triangle, one angle is exactly 90 degrees.
• This triangle has three sides: the base (horizontal part), the height (vertical part), and the hypotenuse (the slanted part).

In our problem:

• The height (vertical leg) is the part of the bamboo still standing.
• The base (horizontal leg) is the distance from the stem to where the top of the bamboo touches the ground.
• The hypotenuse is the length of the bamboo after it broke.

Visualizing the problem as a right triangle makes it easier to apply the Pythagorean theorem.
The right triangle properties are essential since they provide a clear structure to solve the problem.
This sets the foundation for using the significant Pythagorean theorem.

Solving Equations

Solving equations is a step-by-step process to find unknown values. Here, we use the Pythagorean theorem to solve for 'h', the height of the break.
Recall the Pythagorean theorem:
\( a^2 + b^2 = c^2 \)

• a and b are the legs of the right triangle.
• c is the hypotenuse.

Substituting the known values into the theorem:

• Let height 'h' be one leg.
• Let 3 feet be the other leg.
• The hypotenuse is the broken part of the bamboo, which is \( 10 - h \).

So the equation becomes:
\[ h^2 + 3^2 = (10 - h)^2 \]
Expanding the right side:

• \[ h^2 + 9 = 100 - 20h + h^2 \]

Isolating the variable, 'h', we get:

• \[ 9 = 100 - 20h \]
• \[ 9 - 100 = -20h \]
• \[ -91 = -20h \]

Solving for 'h':
\[ h = \frac{91}{20} \]
Which simplifies to:
\[ h = 4.55 \text{ feet} \]
This means the bamboo broke at about 4.55 feet from the ground.

Mathematical Visualization

Visualizing problems can significantly help in understanding and solving them. Here’s how we can apply mathematical visualization to our problem:

• Create a mental or drawn image of the right triangle formed by the broken bamboo.
• Identify each part of the triangle: the vertical leg (remaining bamboo height), the horizontal leg (distance from the stem), and the hypotenuse (total length after the break).

By visualizing the triangle and segments, it becomes easier to see how to apply the Pythagorean theorem.
Visualization tips:

• Draw the right triangle on paper.
• Label all known measurements: height ( \( h \)), base ( \( 3 \) feet), and hypotenuse ( \( 10 - h \)).

This visual aid can simplify the process, helping you set up the equation correctly.
It makes clear why we use the Pythagorean theorem and assists in solving for 'h'.