The base of a certain solid is the triangle with vertices at (−12,6), (6,6), and the origin. Cross-sections perpendicular to the y-axis are squares.
Oops. My bad. I took cross-sections perpendicular to the x-axis.
So see what you can do to fix my mistake, taking horizontal strips as the base of each square.
To find the equation of the base triangle, we'll first find the equation of the line passing through the points (−12,6) and (6,6).
The formula for the equation of a line passing through two points (x₁, y₁) and (x₂, y₂) is:
(y - y₁) = ((y₂ - y₁)/(x₂ - x₁)) * (x - x₁)
Substituting the given values:
(y - 6) = ((6 - 6)/(6 - (-12))) * (x - (-12))
Simplifying, we have:
(y - 6) = 0 * (x + 12)
Therefore, the equation of the line passing through the given points is:
y - 6 = 0
Simplifying further, we find:
y = 6
Since the solid has a triangular base, the shape is a triangular prism.
To find the cross-sectional area of the prism perpendicular to the y-axis, which is a square, we need to determine the length of one side of the square.
Since the cross-section is perpendicular to the y-axis, the side of the square will be the same as the height of the prism.
In this case, the height of the prism is the difference in y-coordinates between the topmost vertex of the triangle and the x-axis (origin).
The topmost vertex of the triangle is at coordinates (0, 6), and the x-axis is at y = 0. Therefore, the height (or side length of the square) is 6 units.
Hence, the cross-sectional area of the prism perpendicular to the y-axis is (side length)^2 = 6^2 = 36 square units.
The area of the solid is a set of vertical strips of width dx.
The other edges of the triangle are the lines y = -x/2 and y=x
So the volume of the solid is
v = ∫[-12,0] (6 + x/2)^2 dx + ∫[0,6] (6-x)^2 dx = 144 + 72 = 216