The base of a certain solid is the triangle with vertices at (-10,5), (5,5), and the origin. Cross-sections perpendicular to the y-axis are squares. What is the volume of the solid?
To find the volume of the solid, we need to first determine the area of the base and then integrate it along the y-axis.
Step 1: Find the area of the base triangle.
To find the area of a triangle given its vertices, we can use the shoelace formula. The formula states that if the vertices of a triangle are (x1, y1), (x2, y2), and (x3, y3), then the area of the triangle is:
Area = |(x1 * y2 + x2 * y3 + x3 * y1 - y1 * x2 - y2 * x3 - y3 * x1)| / 2
Using the given vertices of the triangle, we can calculate the area:
Area = |(-10 * 5 + 5 * 0 + 0 * 5 - 5 * 5 - 0 * 0 - 5 * (-10))| / 2
Area = |-50 - 0 + 0 - 25 - 0 + 50| / 2
Area = |0| / 2
Area = 0
Therefore, the area of the base triangle is 0.
Step 2: Integrate the area of the cross-sections along the y-axis.
Since the cross-sections are squares, their area is simply the side length squared.
To calculate the side length, we need to find the distance between the y-axis and the line containing the vertices (-10,5) and (5,5). Since the line is horizontal, the distance is the difference in x-coordinates.
Distance = |x2 - x1|
Distance = |5 - (-10)|
Distance = |5 + 10|
Distance = 15
The area of each square cross-section is 15^2 = 225.
Now, we need to integrate the area of the cross-sections along the y-axis to find the total volume. Since the cross-sections are squares, the volume element is the area of each cross-section multiplied by a differential length, dy.
V = ∫(Area) dy
V = ∫(225) dy
V = 225 * ∫(dy)
V = 225 * y (evaluated from 0 to 5)
V = 225 * (5 - 0)
V = 225 * 5
V = 1125
Therefore, the volume of the solid is 1125 cubic units.