Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**WebAssign Graph Evaluation: Finding the Centroid**

This exercise involves sketching the region bounded by the curves, identified by the equation \(5x + 2y = 10\), and the lines \(x = 0\) and \(y = 0\). The goal is to visually estimate the location of the centroid of this bounded region.

### Diagrams Overview

The image provides four plots, each representing a right-angled triangular region in the coordinate system with the vertices at \((0, 0)\), \((2, 0)\), and \((0, 5)\). The diagonal of the triangle corresponds to the line \(5x + 2y = 10\). In each plot, a red dot indicates a different possible position for the centroid.

1. **Top Left Plot:**
   - X-axis ranges from 0 to 2.5
   - Y-axis ranges from 0 to 5
   - Red dot positioned approximately at \((0.5, 1.5)\)

2. **Top Right Plot:**
   - X-axis ranges from 0 to 5
   - Y-axis ranges from 0 to 2.5
   - Red dot positioned approximately at \((2, 0.5)\)

3. **Bottom Left Plot:**
   - X-axis ranges from 0 to 5
   - Y-axis ranges from 0 to 2.5
   - Red dot positioned approximately at \((1, 0.5)\)

4. **Bottom Right Plot:**
   - X-axis ranges from 0 to 2.5
   - Y-axis ranges from 0 to 5
   - Red dot positioned approximately at \((0.5, 2)\)

### Task Instructions

- Review each plot to estimate which red dot best represents the centroid of the triangular region.
- Use the interactive option (circles below graphs) to select the plot that you believe accurately identifies the centroid.

### Conclusion

Finally, calculate and enter the exact coordinates of the centroid in the provided form:
\[
(\bar{x}, \bar{y}) = \left( \text{[input field]}, \text{[input field]} \right)
\] 

This exercise enhances understanding of geometric centroids and encourages spatial reasoning and analysis through visual aids.
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Transcribed Image Text:**WebAssign Graph Evaluation: Finding the Centroid** This exercise involves sketching the region bounded by the curves, identified by the equation \(5x + 2y = 10\), and the lines \(x = 0\) and \(y = 0\). The goal is to visually estimate the location of the centroid of this bounded region. ### Diagrams Overview The image provides four plots, each representing a right-angled triangular region in the coordinate system with the vertices at \((0, 0)\), \((2, 0)\), and \((0, 5)\). The diagonal of the triangle corresponds to the line \(5x + 2y = 10\). In each plot, a red dot indicates a different possible position for the centroid. 1. **Top Left Plot:** - X-axis ranges from 0 to 2.5 - Y-axis ranges from 0 to 5 - Red dot positioned approximately at \((0.5, 1.5)\) 2. **Top Right Plot:** - X-axis ranges from 0 to 5 - Y-axis ranges from 0 to 2.5 - Red dot positioned approximately at \((2, 0.5)\) 3. **Bottom Left Plot:** - X-axis ranges from 0 to 5 - Y-axis ranges from 0 to 2.5 - Red dot positioned approximately at \((1, 0.5)\) 4. **Bottom Right Plot:** - X-axis ranges from 0 to 2.5 - Y-axis ranges from 0 to 5 - Red dot positioned approximately at \((0.5, 2)\) ### Task Instructions - Review each plot to estimate which red dot best represents the centroid of the triangular region. - Use the interactive option (circles below graphs) to select the plot that you believe accurately identifies the centroid. ### Conclusion Finally, calculate and enter the exact coordinates of the centroid in the provided form: \[ (\bar{x}, \bar{y}) = \left( \text{[input field]}, \text{[input field]} \right) \] This exercise enhances understanding of geometric centroids and encourages spatial reasoning and analysis through visual aids.
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