Question
A ladder that is 22 feet long is propped up against a 15-foot-tall building. Use the drop downs to answer the following questions.
What is the unknown information that could be found using the Pythagorean theorem?
What measures represent each variable of the Pythagorean theorem, if b is the unknown length?
a =
c =
What is the approximate length of b, rounding to the nearest hundredth?
What is the unknown information that could be found using the Pythagorean theorem?
What measures represent each variable of the Pythagorean theorem, if b is the unknown length?
a =
c =
What is the approximate length of b, rounding to the nearest hundredth?
Answers
GPT-4o mini
To solve this problem, we will use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)). The equation is as follows:
\[
a^2 + b^2 = c^2
\]
In this scenario:
- The length of the ladder (which is the hypotenuse) is 22 feet, so \(c = 22\).
- The height of the building is 15 feet, so we can denote that as \(a = 15\).
- The unknown length \(b\) will represent the distance from the base of the building to the bottom of the ladder (the horizontal distance).
Now we can summarize the variables:
- \(a = 15\) feet (the height of the building)
- \(c = 22\) feet (the length of the ladder)
- \(b\) is the unknown length we need to find.
Now we can apply the Pythagorean theorem to find \(b\):
\[
15^2 + b^2 = 22^2
\]
Calculating the squares:
\[
225 + b^2 = 484
\]
Now, we solve for \(b^2\):
\[
b^2 = 484 - 225
\]
\[
b^2 = 259
\]
Now, taking the square root of both sides to find \(b\):
\[
b = \sqrt{259} \approx 16.093
\]
Rounding to the nearest hundredth, \(b \approx 16.09\) feet.
### Summary:
- **Unknown Length**: \(b\)
- **Values**:
- \(a = 15\) feet
- \(c = 22\) feet
- **Approximate length of \(b\)**: \(16.09\) feet.
\[
a^2 + b^2 = c^2
\]
In this scenario:
- The length of the ladder (which is the hypotenuse) is 22 feet, so \(c = 22\).
- The height of the building is 15 feet, so we can denote that as \(a = 15\).
- The unknown length \(b\) will represent the distance from the base of the building to the bottom of the ladder (the horizontal distance).
Now we can summarize the variables:
- \(a = 15\) feet (the height of the building)
- \(c = 22\) feet (the length of the ladder)
- \(b\) is the unknown length we need to find.
Now we can apply the Pythagorean theorem to find \(b\):
\[
15^2 + b^2 = 22^2
\]
Calculating the squares:
\[
225 + b^2 = 484
\]
Now, we solve for \(b^2\):
\[
b^2 = 484 - 225
\]
\[
b^2 = 259
\]
Now, taking the square root of both sides to find \(b\):
\[
b = \sqrt{259} \approx 16.093
\]
Rounding to the nearest hundredth, \(b \approx 16.09\) feet.
### Summary:
- **Unknown Length**: \(b\)
- **Values**:
- \(a = 15\) feet
- \(c = 22\) feet
- **Approximate length of \(b\)**: \(16.09\) feet.