Final answer:
To find the centroid of the region bounded by the curves 9x + 8y = 72, x = 0, and y = 0, we can graph the equations and use the formula for finding the centroid of a triangle.
Step-by-step explanation:
To find the centroid of the region bounded by the curves, we need to first graph the equations and then calculate the coordinates of the centroid. The given equations are 9x + 8y = 72, x = 0, and y = 0.
First, let's graph the equation 9x + 8y = 72. We can rewrite it in terms of y as y = (72 - 9x) / 8. For the x-intercept, we set y = 0 and solve for x: x = 8. For the y-intercept, we set x = 0 and solve for y: y = 9. So, the graph passes through the points (0, 9) and (8, 0).
Next, let's graph the lines x = 0 and y = 0. These lines represent the x-axis and y-axis respectively. The region bounded by these curves is a triangle.
To find the centroid of the region, we can use the formula:
(x, y) = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)
Where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices of the triangle.
By plugging in the coordinates of the vertices, we can find the exact coordinates of the centroid.
Therefore, the coordinates of the centroid are ((0 + 8 + 0)/3, (9 + 0 + 0)/3) = (8/3, 3).