Question
If you have a 24 foot ladder that is leaning against the building and touching the ground 6 feet from the building. How high up the building is the top of the ladder?
(1 point)
Responses
6^2+b^2=24^2; 24.7 ft
6 squared plus b squared is equal to 24 squared ; 24.7 ft
6^2+b^2=24^2; 23.24 ft
6 squared plus b squared is equal to 24 squared ; 23.24 ft
a^2+24^2=6^2; 23.24 ft
A squared plus 24 squared is equal to 6 squared ; 23.24 ft
24^2+6^2=c^2; 24.7 ft
(1 point)
Responses
6^2+b^2=24^2; 24.7 ft
6 squared plus b squared is equal to 24 squared ; 24.7 ft
6^2+b^2=24^2; 23.24 ft
6 squared plus b squared is equal to 24 squared ; 23.24 ft
a^2+24^2=6^2; 23.24 ft
A squared plus 24 squared is equal to 6 squared ; 23.24 ft
24^2+6^2=c^2; 24.7 ft
Answers
GPT-4o mini
To find out how high the ladder reaches up the building, we can use the Pythagorean theorem. According to the Pythagorean theorem, in a right triangle:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the length of the hypotenuse (the ladder), \( a \) is the height the ladder reaches up the building, and \( b \) is the distance from the base of the ladder to the building.
In this scenario:
- \( c = 24 \) feet (the length of the ladder)
- \( b = 6 \) feet (the distance from the base of the ladder to the building)
We need to find \( a \), the height the ladder reaches up the building. Plugging into the equation, we have:
\[ a^2 + 6^2 = 24^2 \]
Calculating \( 6^2 \) and \( 24^2 \):
\[ 6^2 = 36 \]
\[ 24^2 = 576 \]
Now we can substitute these values back into the equation:
\[ a^2 + 36 = 576 \]
Subtract 36 from both sides:
\[ a^2 = 576 - 36 \]
\[ a^2 = 540 \]
Now, take the square root of both sides to find \( a \):
\[ a = \sqrt{540} \]
\[ a \approx 23.24 \text{ feet} \]
Thus, the height that the top of the ladder reaches on the building is approximately **23.24 feet**.
The correct response is:
**6 squared plus b squared is equal to 24 squared; 23.24 ft**.
\[ a^2 + b^2 = c^2 \]
where \( c \) is the length of the hypotenuse (the ladder), \( a \) is the height the ladder reaches up the building, and \( b \) is the distance from the base of the ladder to the building.
In this scenario:
- \( c = 24 \) feet (the length of the ladder)
- \( b = 6 \) feet (the distance from the base of the ladder to the building)
We need to find \( a \), the height the ladder reaches up the building. Plugging into the equation, we have:
\[ a^2 + 6^2 = 24^2 \]
Calculating \( 6^2 \) and \( 24^2 \):
\[ 6^2 = 36 \]
\[ 24^2 = 576 \]
Now we can substitute these values back into the equation:
\[ a^2 + 36 = 576 \]
Subtract 36 from both sides:
\[ a^2 = 576 - 36 \]
\[ a^2 = 540 \]
Now, take the square root of both sides to find \( a \):
\[ a = \sqrt{540} \]
\[ a \approx 23.24 \text{ feet} \]
Thus, the height that the top of the ladder reaches on the building is approximately **23.24 feet**.
The correct response is:
**6 squared plus b squared is equal to 24 squared; 23.24 ft**.