Answer :
Sure! Let's analyze each part of the question step by step.
(a) Center of Mass Calculation:
For this part, we need to determine if the given center of mass [tex]\(\bar{x} = -9\)[/tex] for the system is correct.
1. Identify Masses and Positions:
- Particles have masses: 3 slugs, 1 slug, and 8 slugs.
- Their positions are: [tex]\(x_1 = -3\)[/tex], [tex]\(x_2 = 1\)[/tex], and [tex]\(x_3 = 7\)[/tex].
2. Calculate the Total Mass:
[tex]\[
\text{Total mass} = 3 + 1 + 8 = 12 \text{ slugs}
\][/tex]
3. Calculate the Center of Mass using the formula:
[tex]\[
\bar{x} = \frac{m_1x_1 + m_2x_2 + m_3x_3}{\text{Total mass}}
\][/tex]
Substituting the given values:
[tex]\[
\bar{x} = \frac{3(-3) + 1(1) + 8(7)}{12}
\][/tex]
[tex]\[
\bar{x} = \frac{-9 + 1 + 56}{12}
\][/tex]
[tex]\[
\bar{x} = \frac{48}{12} = 4
\][/tex]
So, the calculated center of mass [tex]\(\bar{x} = 4\)[/tex]. This tells us that the statement is false because the center of mass is given incorrectly as [tex]\(-9\)[/tex].
(b) Surface Area of Revolution:
For part (b), we're dealing with a curve [tex]\(y = f(x)\)[/tex] on the interval [tex]\([1,4]\)[/tex] rotated around the y-axis.
- The formula given in the statement is:
[tex]\[
2\pi \int_1^4 f(x) \sqrt{1 + (f'(x))^2} \, dx
\][/tex]
This formula is incorrect for finding the surface area when revolving around the y-axis. The correct formula should be:
[tex]\[
2\pi \int_1^4 x \cdot \sqrt{1 + (f'(x))^2} \, dx
\][/tex]
Therefore, the statement is false because it incorrectly states the formula for the surface area of revolution.
In conclusion:
- Part (a) is false because the correct center of mass is [tex]\(4\)[/tex], not [tex]\(-9\)[/tex].
- Part (b) is false because the correct formula for surface area when revolving around the y-axis involves an integral multiplied by [tex]\(x\)[/tex].
(a) Center of Mass Calculation:
For this part, we need to determine if the given center of mass [tex]\(\bar{x} = -9\)[/tex] for the system is correct.
1. Identify Masses and Positions:
- Particles have masses: 3 slugs, 1 slug, and 8 slugs.
- Their positions are: [tex]\(x_1 = -3\)[/tex], [tex]\(x_2 = 1\)[/tex], and [tex]\(x_3 = 7\)[/tex].
2. Calculate the Total Mass:
[tex]\[
\text{Total mass} = 3 + 1 + 8 = 12 \text{ slugs}
\][/tex]
3. Calculate the Center of Mass using the formula:
[tex]\[
\bar{x} = \frac{m_1x_1 + m_2x_2 + m_3x_3}{\text{Total mass}}
\][/tex]
Substituting the given values:
[tex]\[
\bar{x} = \frac{3(-3) + 1(1) + 8(7)}{12}
\][/tex]
[tex]\[
\bar{x} = \frac{-9 + 1 + 56}{12}
\][/tex]
[tex]\[
\bar{x} = \frac{48}{12} = 4
\][/tex]
So, the calculated center of mass [tex]\(\bar{x} = 4\)[/tex]. This tells us that the statement is false because the center of mass is given incorrectly as [tex]\(-9\)[/tex].
(b) Surface Area of Revolution:
For part (b), we're dealing with a curve [tex]\(y = f(x)\)[/tex] on the interval [tex]\([1,4]\)[/tex] rotated around the y-axis.
- The formula given in the statement is:
[tex]\[
2\pi \int_1^4 f(x) \sqrt{1 + (f'(x))^2} \, dx
\][/tex]
This formula is incorrect for finding the surface area when revolving around the y-axis. The correct formula should be:
[tex]\[
2\pi \int_1^4 x \cdot \sqrt{1 + (f'(x))^2} \, dx
\][/tex]
Therefore, the statement is false because it incorrectly states the formula for the surface area of revolution.
In conclusion:
- Part (a) is false because the correct center of mass is [tex]\(4\)[/tex], not [tex]\(-9\)[/tex].
- Part (b) is false because the correct formula for surface area when revolving around the y-axis involves an integral multiplied by [tex]\(x\)[/tex].
