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 sidesStep 3: 4(1437)3π=r3 ---> Multiply by 4 and Divide by 3 on both sidesStep 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)ResponsesStep 1 used the wrong formula for a sphereStep 1 used the wrong formula for a sphereStep 2 should have multiplied by π on both sidesStep 2 should have multiplied by pi on both sidesStep 3 should have multiplied by 3 and divided by 4 on both sidesStep 3 should have multiplied by 3 and divided by 4 on both sidesStep 4 should have taken the cube root instead of dividing by 3Step 4 should have taken the cube root instead of dividing by 3Step 4 should have taken the square root instead of dividing by 3Step 4 should have taken the square root instead of dividing by 3
Answers
GPT-4o mini
Let's analyze the steps provided and identify the mistakes:
1. **Step 1**: The formula for the volume of a sphere is indeed \( V = \frac{4}{3} \pi r^3 \). This step correctly uses the right formula, so there is no mistake here.
2. **Step 2**: The transition from \( 1437 = \frac{4}{3} \pi r^3 \) to \( 1437\pi = \frac{4}{3}r^3 \) is incorrect. The formula should actually be \( \frac{4}{3} \pi r^3 = 1437 \). To isolate \( r^3 \), we would multiply both sides by \( \frac{3}{4} \) rather than multiplying both sides by \( \pi \).
3. **Step 3**: This step claims to have multiplied by 4 and divided by 3, which is also incorrect. Correctly, it should be: \( r^3 = \frac{3 \cdot 1437}{4\pi} \).
4. **Step 4**: The error here is assuming that you should divide by 3. Instead, to find \( r \) from \( r^3 \), you need to take the cube root.
Summarizing the mistakes:
1. **Step 2 should have multiplied by π on both sides** - Instead, it should have been isolated correctly by multiplying both sides by \( \frac{3}{4} \) after correctly setting the equation.
2. **Step 4 should have taken the cube root instead of dividing by 3** - The correct step is to take the cube root of \( r^3 \) to find \( r \).
Thus, the correct mistakes are:
- Step 2 should have multiplied by \( \frac{3}{4} \) on both sides.
- Step 4 should have taken the cube root instead of dividing by 3.
1. **Step 1**: The formula for the volume of a sphere is indeed \( V = \frac{4}{3} \pi r^3 \). This step correctly uses the right formula, so there is no mistake here.
2. **Step 2**: The transition from \( 1437 = \frac{4}{3} \pi r^3 \) to \( 1437\pi = \frac{4}{3}r^3 \) is incorrect. The formula should actually be \( \frac{4}{3} \pi r^3 = 1437 \). To isolate \( r^3 \), we would multiply both sides by \( \frac{3}{4} \) rather than multiplying both sides by \( \pi \).
3. **Step 3**: This step claims to have multiplied by 4 and divided by 3, which is also incorrect. Correctly, it should be: \( r^3 = \frac{3 \cdot 1437}{4\pi} \).
4. **Step 4**: The error here is assuming that you should divide by 3. Instead, to find \( r \) from \( r^3 \), you need to take the cube root.
Summarizing the mistakes:
1. **Step 2 should have multiplied by π on both sides** - Instead, it should have been isolated correctly by multiplying both sides by \( \frac{3}{4} \) after correctly setting the equation.
2. **Step 4 should have taken the cube root instead of dividing by 3** - The correct step is to take the cube root of \( r^3 \) to find \( r \).
Thus, the correct mistakes are:
- Step 2 should have multiplied by \( \frac{3}{4} \) on both sides.
- Step 4 should have taken the cube root instead of dividing by 3.