Question
If the volume of a spherical ball is 1437 cubic inches, what is the radius?
Step 1: V=43πr3
---> Substitute in 1437 for V and solve for r.
Step 2: 1437π=43r3
---> Divide by π
on both sides
Step 3: 4(1437)3π=r3
---> Multiply by 4 and Divide by 3 on both sides
Step 4: 4(1437)3⋅3π=r
---> Divide by 3 on both sides
There are 2 mistakes in this process. Choose the correct 2 mistakes.
(2 points)
Responses
Step 1 used the wrong formula for a sphere
Step 1 used the wrong formula for a sphere
Step 2 should have multiplied by π
on both sides
Step 2 should have multiplied by pi on both sides
Step 3 should have multiplied by 3 and divided by 4 on both sides
Step 3 should have multiplied by 3 and divided by 4 on both sides
Step 4 should have taken the cube root instead of dividing by 3
Step 4 should have taken the cube root instead of dividing by 3
Step 4 should have taken the square root instead of dividing by 3
Step 4 should have taken the square root instead of dividing by 3
Skip to navigation
Step 1: V=43πr3
---> Substitute in 1437 for V and solve for r.
Step 2: 1437π=43r3
---> Divide by π
on both sides
Step 3: 4(1437)3π=r3
---> Multiply by 4 and Divide by 3 on both sides
Step 4: 4(1437)3⋅3π=r
---> Divide by 3 on both sides
There are 2 mistakes in this process. Choose the correct 2 mistakes.
(2 points)
Responses
Step 1 used the wrong formula for a sphere
Step 1 used the wrong formula for a sphere
Step 2 should have multiplied by π
on both sides
Step 2 should have multiplied by pi on both sides
Step 3 should have multiplied by 3 and divided by 4 on both sides
Step 3 should have multiplied by 3 and divided by 4 on both sides
Step 4 should have taken the cube root instead of dividing by 3
Step 4 should have taken the cube root instead of dividing by 3
Step 4 should have taken the square root instead of dividing by 3
Step 4 should have taken the square root instead of dividing by 3
Skip to navigation
Answers
GPT-4o mini
Let's analyze the steps provided to identify the mistakes.
1. **Step 1** states:
\[
V = \frac{4}{3}\pi r^3
\]
This is the correct formula for the volume of a sphere, so there is no mistake in Step 1.
2. **Step 2** states:
\[
1437\pi = \frac{4}{3}r^3
\]
Here it should actually be:
\[
1437 = \frac{4}{3}\pi r^3
\]
To isolate \( r^3 \), we would need to multiply by \( \frac{3}{4} \) and divide by \( \pi \) correctly. The step suggests dividing by \( \pi \), which is correct, but the subsequent equation is incorrect. So, there is an error.
3. **Step 3** attempts to manipulate the equation, but the formula for \( r^3 \) should come after properly isolating \( r^3 \) from the previous step. Based on the correct equation of Step 2, the manipulation in Step 3 does not follow logically from Step 2.
4. **Step 4** suggests dividing by 3. Instead, once \( r^3 \) is isolated correctly, we should take the cube root, not divide. This is a mistake.
**Summarizing the mistakes:**
- **Step 1** is correctly using the volume formula.
- **Step 2** correctly indicates that you should multiply by \( \frac{3}{4} \), but it incorrectly shows the equation.
- **Step 3** indicates misunderstandings in working out \( r^3 \).
- **Step 4** should have taken the cube root instead of dividing.
Based on this analysis, the two mistakes are:
- **Step 2 should have multiplied by \( \frac{3}{4} \) instead of confusing the manipulation.**
- **Step 4 should have taken the cube root instead of dividing by 3.**
So the correct responses would be:
1. **Step 2 should have multiplied by 3 and divided by 4 on both sides.**
2. **Step 4 should have taken the cube root instead of dividing by 3.**
1. **Step 1** states:
\[
V = \frac{4}{3}\pi r^3
\]
This is the correct formula for the volume of a sphere, so there is no mistake in Step 1.
2. **Step 2** states:
\[
1437\pi = \frac{4}{3}r^3
\]
Here it should actually be:
\[
1437 = \frac{4}{3}\pi r^3
\]
To isolate \( r^3 \), we would need to multiply by \( \frac{3}{4} \) and divide by \( \pi \) correctly. The step suggests dividing by \( \pi \), which is correct, but the subsequent equation is incorrect. So, there is an error.
3. **Step 3** attempts to manipulate the equation, but the formula for \( r^3 \) should come after properly isolating \( r^3 \) from the previous step. Based on the correct equation of Step 2, the manipulation in Step 3 does not follow logically from Step 2.
4. **Step 4** suggests dividing by 3. Instead, once \( r^3 \) is isolated correctly, we should take the cube root, not divide. This is a mistake.
**Summarizing the mistakes:**
- **Step 1** is correctly using the volume formula.
- **Step 2** correctly indicates that you should multiply by \( \frac{3}{4} \), but it incorrectly shows the equation.
- **Step 3** indicates misunderstandings in working out \( r^3 \).
- **Step 4** should have taken the cube root instead of dividing.
Based on this analysis, the two mistakes are:
- **Step 2 should have multiplied by \( \frac{3}{4} \) instead of confusing the manipulation.**
- **Step 4 should have taken the cube root instead of dividing by 3.**
So the correct responses would be:
1. **Step 2 should have multiplied by 3 and divided by 4 on both sides.**
2. **Step 4 should have taken the cube root instead of dividing by 3.**