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Question

An 18-foot ladder is leaning against a wall. The distance on the ground from the ladder to the wall is 8 feet. How high up is the ladder on the wall?
Steps for solving real-world problem:

Traingle: 18 ft, b, 8 ft

1. Create a diagram.
2. Determine variable values.
3. Substitute into the Pythagorean theorem.

a = 8 ft, b = ?, c = _ ft
a^(2) + b^(2) = c^(2)
_^(2) + b^(2) = _^(2)

Answer 1

To solve the problem, we will follow the steps you've laid out using the Pythagorean theorem.

1. Create a diagram:

   • You have a right triangle where one leg (the distance from the wall) is 8 feet (let's call this leg "a"), the other leg (the height on the wall) is unknown (let's call this leg "b"), and the hypotenuse (the length of the ladder) is 18 feet (let's call this "c").

2. Determine variable values:

   • a = 8 ft (distance from the wall)
   • b = ? (height up the wall)
   • c = 18 ft (length of the ladder)

3. Substitute into the Pythagorean theorem:

   • The Pythagorean theorem states that \( a^{2} + b^{2} = c^{2} \).

   Now you can substitute the known values into the equation: \[ 8^{2} + b^{2} = 18^{2} \] Calculate the squares: \[ 64 + b^{2} = 324 \]

   Now, solve for \( b^{2} \): \[ b^{2} = 324 - 64 \] \[ b^{2} = 260 \]

4. Finally, take the square root to find \( b \): \[ b = \sqrt{260} \approx 16.12 \text{ feet} \]

So, the height up the wall where the ladder reaches is approximately 16.12 feet.