Question
If the ladder is 10 m long and 5 m from the base of the building, how high up the building will the ladder reach?
(1 point)
Responses
52+x2=102
; x = √75 m, which is an irrational number and would be estimated to be 8.7 m.
5 squared plus x squared is equal to 10 squared ; x = √75 m, which is an irrational number and would be estimated to be 8.7 m.
102+52=c2
; c = 125 m, which is a rational, whole number
10 squared plus 5 squared is equal to c squared ; c = 125 m, which is a rational, whole number
102−−−√+52=c2
; c = √125, which is an irrational number and must be estimated at 11.2 m.
square root of 10 squared end root plus 5 squared is equal to c squared ; c = √125, which is an irrational number and must be estimated at 11.2 m.
52+x2=102
; x = 75 m, which is a rational, whole number
Answers
GPT-4o mini
To find out how high the ladder reaches up the building, we can use the Pythagorean theorem. In this scenario:
- The ladder represents the hypotenuse of the right triangle (10 m).
- The distance from the base of the building to the foot of the ladder represents one leg of the triangle (5 m).
- We need to find the height the ladder reaches up the building, which we can denote as \(x\) (the other leg of the triangle).
According to the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
Where:
- \(a\) = height of the ladder up the wall (x)
- \(b\) = distance from the wall to the base of the ladder (5 m)
- \(c\) = length of the ladder (10 m)
Substituting these values into the equation gives us:
\[
x^2 + 5^2 = 10^2
\]
This simplifies to:
\[
x^2 + 25 = 100
\]
Now, subtracting 25 from both sides:
\[
x^2 = 75
\]
Taking the square root of both sides:
\[
x = \sqrt{75}
\]
This can be simplified further:
\[
x = \sqrt{25 \cdot 3} = 5\sqrt{3} \approx 8.66 \text{ m}
\]
Thus, the ladder will reach approximately 8.7 m up the building.
Now, looking at the responses provided:
The correct answer is the first option:
"5 squared plus x squared is equal to 10 squared ; x = √75 m, which is an irrational number and would be estimated to be 8.7 m."
- The ladder represents the hypotenuse of the right triangle (10 m).
- The distance from the base of the building to the foot of the ladder represents one leg of the triangle (5 m).
- We need to find the height the ladder reaches up the building, which we can denote as \(x\) (the other leg of the triangle).
According to the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
Where:
- \(a\) = height of the ladder up the wall (x)
- \(b\) = distance from the wall to the base of the ladder (5 m)
- \(c\) = length of the ladder (10 m)
Substituting these values into the equation gives us:
\[
x^2 + 5^2 = 10^2
\]
This simplifies to:
\[
x^2 + 25 = 100
\]
Now, subtracting 25 from both sides:
\[
x^2 = 75
\]
Taking the square root of both sides:
\[
x = \sqrt{75}
\]
This can be simplified further:
\[
x = \sqrt{25 \cdot 3} = 5\sqrt{3} \approx 8.66 \text{ m}
\]
Thus, the ladder will reach approximately 8.7 m up the building.
Now, looking at the responses provided:
The correct answer is the first option:
"5 squared plus x squared is equal to 10 squared ; x = √75 m, which is an irrational number and would be estimated to be 8.7 m."