Question
If the length of the ramp is 3 feet long and the vertical rise is 20 inches, explain why it is or is not possible to have a horizontal distance of 16 inches.
(5 points)
Responses
It is possible because 3 feet is 39 inches.
It is possible because 3 feet is 39 inches.
It is possible because 3 feet is 36 inches and a horizontal distance of 16 inches is between 20 and 36 inches.
It is possible because 3 feet is 36 inches and a horizontal distance of 16 inches is between 20 and 36 inches.
It is not possible because 16 inches is not more than the difference between 36 and 20 inches.
It is not possible because 16 inches is not more than the difference between 36 and 20 inches.
It is not possible because the length of the ramp is longer than the vertical rise.
It is not possible because the length of the ramp is longer than the vertical rise.
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Answers
GPT-4o mini
To determine if it is possible to have a horizontal distance of 16 inches given that the ramp is 3 feet long (39 inches) and has a vertical rise of 20 inches, we employ the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle:
\[
a^2 + b^2 = c^2
\]
where \( a \) is the horizontal distance, \( b \) is the vertical rise, and \( c \) is the length of the ramp.
Here:
- \( b = 20 \) inches (vertical rise)
- \( c = 39 \) inches (length of the ramp)
- \( a = 16 \) inches (horizontal distance we want to check)
We can rearrange the equation to find \( a^2 \):
\[
a^2 = c^2 - b^2
\]
Substituting the known values:
\[
a^2 = 39^2 - 20^2
\]
\[
a^2 = 1521 - 400
\]
\[
a^2 = 1121
\]
Now, we can take the square root to find \( a \):
\[
a \approx \sqrt{1121} \approx 33.5 \text{ inches}
\]
Since 33.5 inches is much greater than 16 inches, the implication is that a horizontal distance of 16 inches could be possible since it is less than that calculated maximum distance.
Now looking at the provided responses:
1. "It is possible because 3 feet is 39 inches." - This is partially true, but does not fully support or explain the conclusion.
2. "It is possible because 3 feet is 36 inches and a horizontal distance of 16 inches is between 20 and 36 inches." - This is incorrect, as 3 feet is not 36 inches.
3. "It is not possible because 16 inches is not more than the difference between 36 and 20 inches." - This is incorrect based on our previous calculations.
4. "It is not possible because the length of the ramp is longer than the vertical rise." - This is also misleading, as it doesn't directly relate to the feasibility of a horizontal distance of 16 inches.
Given the calculations, the correct understanding is that it **is possible** to have a horizontal distance of 16 inches because this value falls below the maximum possible horizontal distance derived from the calculations, which is approximately 33.5 inches. Thus, none of the provided responses accurately capture this conclusion.
The Pythagorean theorem states that in a right triangle:
\[
a^2 + b^2 = c^2
\]
where \( a \) is the horizontal distance, \( b \) is the vertical rise, and \( c \) is the length of the ramp.
Here:
- \( b = 20 \) inches (vertical rise)
- \( c = 39 \) inches (length of the ramp)
- \( a = 16 \) inches (horizontal distance we want to check)
We can rearrange the equation to find \( a^2 \):
\[
a^2 = c^2 - b^2
\]
Substituting the known values:
\[
a^2 = 39^2 - 20^2
\]
\[
a^2 = 1521 - 400
\]
\[
a^2 = 1121
\]
Now, we can take the square root to find \( a \):
\[
a \approx \sqrt{1121} \approx 33.5 \text{ inches}
\]
Since 33.5 inches is much greater than 16 inches, the implication is that a horizontal distance of 16 inches could be possible since it is less than that calculated maximum distance.
Now looking at the provided responses:
1. "It is possible because 3 feet is 39 inches." - This is partially true, but does not fully support or explain the conclusion.
2. "It is possible because 3 feet is 36 inches and a horizontal distance of 16 inches is between 20 and 36 inches." - This is incorrect, as 3 feet is not 36 inches.
3. "It is not possible because 16 inches is not more than the difference between 36 and 20 inches." - This is incorrect based on our previous calculations.
4. "It is not possible because the length of the ramp is longer than the vertical rise." - This is also misleading, as it doesn't directly relate to the feasibility of a horizontal distance of 16 inches.
Given the calculations, the correct understanding is that it **is possible** to have a horizontal distance of 16 inches because this value falls below the maximum possible horizontal distance derived from the calculations, which is approximately 33.5 inches. Thus, none of the provided responses accurately capture this conclusion.