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A thin 16​-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of​ g/cm. The mass is the area under the density curve. 
a. Find the mass of the left half of the rod ​(00less than or equals≤xless than or equals≤8​).
b. Find the mass of the right half of the rod ​(8less than or equals≤xless than or equals≤16​).
c. Find the mass of the entire rod ​(00less than or equals≤xless than or equals≤16​).
d. Estimate the point along the rod at which it will balance​ (called the center of​ mass).

To solve this problem, we'll approach each question individually, using the given graph where density (in g/cm) is plotted against the rod’s length (in cm). The mass of each section of the rod corresponds to the area under the density curve in that section.

1. **Find the Mass of the Left Half of the Rod** \((0 \leq x \leq 8)\):

   This region includes two parts:
   - From \(x = 0\) to \(x = 2\), the density is approximately constant at \(1 \, \text{g/cm}\).
   - From \(x = 2\) to \(x = 8\), the density linearly increases from \(1 \, \text{g/cm}\) to \(10 \, \text{g/cm}\).

   We can calculate the mass as follows:
   - For \(0 \leq x \leq 2\): The area of this rectangle is \( \text{width} \times \text{height} = 2 \times 1 = 2 \, \text{g}\).
   - For \(2 \leq x \leq 8\): The area of this trapezoid is given by:
     \[
     \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height} = \frac{1}{2} \times (1 + 10) \times 6 = \frac{1}{2} \times 11 \times 6 = 33 \, \text{g}.
     \]
   
   Therefore, the mass of the left half is:
   \[
   2 + 33 = 35 \, \text{g}.
   \]

2. **Find the Mass of the Right Half of the Rod** \((8 \leq x \leq 16)\):

   This region shows a constant density of \(10 \, \text{g/cm}\) from \(x = 8\) to \(x = 16\).
   - The area of this rectangle is \( \text{width} \times \text{height} = 8 \times 10 = 80 \, \text{g}\).

   Thus, the mass of the right half is \(80 \, \text{g}\).

3. **Find the Mass of the Entire Rod** \((0 \leq x \leq 16)\):

   To find the total mass, we add the masses of the left and right halves:
   \[
   35 + 80 = 115 \, \text{g}.
   \]

4. **Estimate the Center of Mass**:

   The center of mass \( \bar{x} \) is the point along the length where the rod balances, given by:
   \[
   \bar{x} = \frac{\int x \cdot \text{density}(x) \, dx}{\text{Total Mass}}.
   \]
   However, since we have discrete areas, we can estimate the center of mass by calculating weighted averages for each segment:

   - For \(0 \leq x \leq 2\), with mass \(2\) g centered around \(x = 1\).
   - For \(2 \leq x \leq 8\), with mass \(33\) g centered around the midpoint \(x = 5\).
   - For \(8 \leq x \leq 16\), with mass \(80\) g centered around \(x = 12\).

   Using these values, we approximate \(\bar{x}\):
   \[
   \bar{x} = \frac{(2 \cdot 1) + (33 \cdot 5) + (80 \cdot 12)}{115} = \frac{2 + 165 + 960}{115} = \frac{1127}{115} \approx 9.8 \, \text{cm}.
   \]

### Summary of Results
- **Mass of left half**: \(35 \, \text{g}\)
- **Mass of right half**: \(80 \, \text{g}\)
- **Total mass**: \(115 \, \text{g}\)
- **Center of mass**: approximately \(9.8 \, \text{cm}\)

Would you like more details or explanations on any part? 

Here are related questions to explore further:
1. How would you calculate the center of mass for a continuous density function?
2. How does the density variation affect the balance point of the rod?
3. What are other methods to approximate the mass from a density graph?
4. How could we verify our results with integral calculus?
5. How would the results change if the rod's density distribution were exponential?

**Tip**: For non-uniform objects, the center of mass shifts toward denser regions.

A thin 16​-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of​ g/cm. The mass is the area under the density curve.

a. Find the mass of the left half of the rod (0 ≤ x ≤ 8).

b. Find the mass of the right half of the rod (8 ≤ x ≤ 16).

c. Find the mass of the entire rod (0 ≤ x ≤ 16).

d. Estimate the point along the rod at which it will balance​ (called the center of mass).

Solve for the mass and center of mass of a 16-cm alloy rod with a non-uniform density. The solution covers calculating the mass of the left and right halves, total mass, and estimating the balance point using integral calculus.

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