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 a rational, whole number
5 squared plus x squared is equal to 10 squared ; x = 75 m, which is a rational, whole number
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 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.
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Answers
GPT-4o mini
To solve the problem, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs (a and b) equals the square of the hypotenuse (c):
\[
c^2 = a^2 + b^2
\]
In this case:
- The length of the ladder is the hypotenuse (\(c\)) and is 10 m.
- The distance from the base of the building (\(a\)) is 5 m.
- The height that the ladder reaches on the building (\(b\)) is what we want to find.
Using the Pythagorean theorem:
\[
10^2 = 5^2 + b^2
\]
Calculating each side:
\[
100 = 25 + b^2
\]
Now, subtract 25 from both sides:
\[
100 - 25 = b^2
\]
\[
75 = b^2
\]
Taking the square root of both sides gives:
\[
b = \sqrt{75}
\]
This can be simplified further:
\[
b = \sqrt{25 \times 3} = 5\sqrt{3} \approx 8.66 \text{ m}
\]
Thus, the height that the ladder will reach up the building is approximately **8.7 m** (rounded to one decimal place).
From the responses provided:
- "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."
This response correctly describes the situation and gives the right height that the ladder will reach.
\[
c^2 = a^2 + b^2
\]
In this case:
- The length of the ladder is the hypotenuse (\(c\)) and is 10 m.
- The distance from the base of the building (\(a\)) is 5 m.
- The height that the ladder reaches on the building (\(b\)) is what we want to find.
Using the Pythagorean theorem:
\[
10^2 = 5^2 + b^2
\]
Calculating each side:
\[
100 = 25 + b^2
\]
Now, subtract 25 from both sides:
\[
100 - 25 = b^2
\]
\[
75 = b^2
\]
Taking the square root of both sides gives:
\[
b = \sqrt{75}
\]
This can be simplified further:
\[
b = \sqrt{25 \times 3} = 5\sqrt{3} \approx 8.66 \text{ m}
\]
Thus, the height that the ladder will reach up the building is approximately **8.7 m** (rounded to one decimal place).
From the responses provided:
- "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."
This response correctly describes the situation and gives the right height that the ladder will reach.