Question
Johnny Lumberjack uses a mallet and wedge to help him split wood. The picture shows the mallet and wedge.
wedge is 8 cm wide and 16 cm long
Johnny Lumberjack uses the wedge because it is a simple machine that provides a mechanical advantage. What is the mechanical advantage of the wedge?
Mechanical Advantage= length of sloped side/width of wedge
a
.5
b
4.5
c
2.0
d
8.0
wedge is 8 cm wide and 16 cm long
Johnny Lumberjack uses the wedge because it is a simple machine that provides a mechanical advantage. What is the mechanical advantage of the wedge?
Mechanical Advantage= length of sloped side/width of wedge
a
.5
b
4.5
c
2.0
d
8.0
Answers
GPT-4o mini
To calculate the mechanical advantage (MA) of the wedge, we use the formula:
\[
\text{Mechanical Advantage} = \frac{\text{length of sloped side}}{\text{width of wedge}}
\]
In this case, the width of the wedge is given as 8 cm. To find the length of the sloped side, we can use the Pythagorean theorem. The wedge can be represented as a right triangle where the width is one leg and the length is the other leg.
Let’s assume:
- The length of the wedge (the height of the triangle) is 16 cm.
- The width of the wedge is 8 cm.
To find the length of the sloped side (hypotenuse), we can use the formula for the hypotenuse of a right triangle:
\[
\text{hypotenuse} = \sqrt{(\text{length})^2 + (\text{width})^2}
\]
\[
\text{hypotenuse} = \sqrt{(16)^2 + (8)^2} = \sqrt{256 + 64} = \sqrt{320} \approx 17.89 \text{ cm}
\]
Now we can calculate the mechanical advantage:
\[
\text{Mechanical Advantage} = \frac{17.89 \text{ cm}}{8 \text{ cm}} \approx 2.24
\]
Based on the provided answer options, none of them exactly match this calculation. However, if we consider the length of the wedge to be the total length (16 cm), not considering the sloped side for the calculation, which might have been implied, we can instead calculate as follows:
If we directly use:
\[
\text{Mechanical Advantage} = \frac{\text{length of wedge}}{\text{width of wedge}} = \frac{16 \text{ cm}}{8 \text{ cm}} = 2 \text{ (option c)}
\]
So the closest option corresponding to the calculation in a simplistically assumed manner is:
**c. 2.0**
\[
\text{Mechanical Advantage} = \frac{\text{length of sloped side}}{\text{width of wedge}}
\]
In this case, the width of the wedge is given as 8 cm. To find the length of the sloped side, we can use the Pythagorean theorem. The wedge can be represented as a right triangle where the width is one leg and the length is the other leg.
Let’s assume:
- The length of the wedge (the height of the triangle) is 16 cm.
- The width of the wedge is 8 cm.
To find the length of the sloped side (hypotenuse), we can use the formula for the hypotenuse of a right triangle:
\[
\text{hypotenuse} = \sqrt{(\text{length})^2 + (\text{width})^2}
\]
\[
\text{hypotenuse} = \sqrt{(16)^2 + (8)^2} = \sqrt{256 + 64} = \sqrt{320} \approx 17.89 \text{ cm}
\]
Now we can calculate the mechanical advantage:
\[
\text{Mechanical Advantage} = \frac{17.89 \text{ cm}}{8 \text{ cm}} \approx 2.24
\]
Based on the provided answer options, none of them exactly match this calculation. However, if we consider the length of the wedge to be the total length (16 cm), not considering the sloped side for the calculation, which might have been implied, we can instead calculate as follows:
If we directly use:
\[
\text{Mechanical Advantage} = \frac{\text{length of wedge}}{\text{width of wedge}} = \frac{16 \text{ cm}}{8 \text{ cm}} = 2 \text{ (option c)}
\]
So the closest option corresponding to the calculation in a simplistically assumed manner is:
**c. 2.0**