Question
if the volume of a spherical ball is 1437 cubic inches, what is the radius?
step 1: V = 4/3pi r^3 --> substitute in 1437 for V and solve for r.
step 2: 1437/pi = 4/3r^3 --> divided by pi on both sides
step 3: 4(1437)/3pi =r^3 --> Multiply by four and divide by 3 on both sides
step 4: 4(1437)/3.3pi = r --> Divide by 3 on both sides
There are 2 mistakes in the process. choose the correct 2 mistakes
step 1: V = 4/3pi r^3 --> substitute in 1437 for V and solve for r.
step 2: 1437/pi = 4/3r^3 --> divided by pi on both sides
step 3: 4(1437)/3pi =r^3 --> Multiply by four and divide by 3 on both sides
step 4: 4(1437)/3.3pi = r --> Divide by 3 on both sides
There are 2 mistakes in the process. choose the correct 2 mistakes
Answers
GPT-4o mini
Let's go through the process step by step to identify the mistakes.
1. **Step 1:** You correctly stated the formula for the volume of a sphere:
\[
V = \frac{4}{3}\pi r^3
\]
You substituted \(V = 1437\):
\[
1437 = \frac{4}{3}\pi r^3
\]
**This step is correct.**
2. **Step 2:** You rearranged the formula to isolate \(r^3\):
\[
1437/\pi = \frac{4}{3}r^3
\]
**This can be simplified but it's mostly okay. Just ensure you are careful with fractions later on. A small mistake here could be misusing the equation.**
3. **Step 3:** You stated:
\[
\frac{4(1437)}{3\pi} = r^3
\]
This step is incorrect. The correct process should divide both sides by \(\frac{4}{3}\). To do this, you would multiply both sides by \(\frac{3}{4}\):
\[
r^3 = \frac{3 \cdot 1437}{4\pi}
\]
**This means Step 3 contains the first mistake.**
4. **Step 4:** You wrote:
\[
r = \frac{4(1437)}{3.3\pi}
\]
This step is also incorrect because it misrepresents the relationship achieved in the previous step. The correct expression after obtaining \(r^3\) should be calculated as follows:
\[
r = \sqrt[3]{\frac{3 \cdot 1437}{4\pi}}
\]
So the second mistake is incorrectly manipulating \(r^3\) to find \(r\).
### The Mistakes:
1. Misrepresenting the fractional relationship in **Step 3** when isolating \(r^3\).
2. Incorrectly calculating \(r\) in **Step 4** from \(r^3\) and not taking the cube root correctly.
### Conclusion:
To find the radius \(r\) correctly after your corrections, you need to take the cube root from \(r^3\):
\[
r = \sqrt[3]{\frac{3 \cdot 1437}{4\pi}}
\]
You can compute that numerical value for the radius.
1. **Step 1:** You correctly stated the formula for the volume of a sphere:
\[
V = \frac{4}{3}\pi r^3
\]
You substituted \(V = 1437\):
\[
1437 = \frac{4}{3}\pi r^3
\]
**This step is correct.**
2. **Step 2:** You rearranged the formula to isolate \(r^3\):
\[
1437/\pi = \frac{4}{3}r^3
\]
**This can be simplified but it's mostly okay. Just ensure you are careful with fractions later on. A small mistake here could be misusing the equation.**
3. **Step 3:** You stated:
\[
\frac{4(1437)}{3\pi} = r^3
\]
This step is incorrect. The correct process should divide both sides by \(\frac{4}{3}\). To do this, you would multiply both sides by \(\frac{3}{4}\):
\[
r^3 = \frac{3 \cdot 1437}{4\pi}
\]
**This means Step 3 contains the first mistake.**
4. **Step 4:** You wrote:
\[
r = \frac{4(1437)}{3.3\pi}
\]
This step is also incorrect because it misrepresents the relationship achieved in the previous step. The correct expression after obtaining \(r^3\) should be calculated as follows:
\[
r = \sqrt[3]{\frac{3 \cdot 1437}{4\pi}}
\]
So the second mistake is incorrectly manipulating \(r^3\) to find \(r\).
### The Mistakes:
1. Misrepresenting the fractional relationship in **Step 3** when isolating \(r^3\).
2. Incorrectly calculating \(r\) in **Step 4** from \(r^3\) and not taking the cube root correctly.
### Conclusion:
To find the radius \(r\) correctly after your corrections, you need to take the cube root from \(r^3\):
\[
r = \sqrt[3]{\frac{3 \cdot 1437}{4\pi}}
\]
You can compute that numerical value for the radius.