The base of a certain solid is the triangle with vertices at (-14,7), (7,7), and the origin. Cross-sections perpendicular to the y-axis are squares.
What is the volume?
Honestly, I can't envisage a solid in just two dimensions.
To find the volume of the solid, we can break it down into an infinite number of infinitesimally thin "slices" perpendicular to the y-axis. Each of these slices will have the shape of a square.
First, let's find the base of the solid, which is the triangle with vertices (-14,7), (7,7), and the origin. To calculate the base area, we can use the formula for the area of a triangle.
The base of the triangle can be found by calculating the distance between the points (-14,7) and (7,7). Since the y-coordinate is the same for both points, the length of the base is 7 - (-14) = 21 units.
To find the height of the triangle, we need to calculate the distance between the origin and the line parallel to the x-axis passing through the point (-14, 7). Since the y-coordinate is 7 for both points, the height of the triangle is 7 units.
Now, we can find the area of the triangle using the formula for the area of a triangle: (base * height) / 2.
Area = (21 * 7) / 2 = 147 / 2 = 73.5 square units.
Since the cross-sections perpendicular to the y-axis are squares, the height of each square is the same as the width of the triangle's base, which is 21 units.
To calculate the volume, we need to integrate the area of each slice along the y-axis from y = 0 (the origin) to y = 7.
V = ∫[0,7] A(y)dy, where A(y) is the area of each slice at height y.
Since the area of each slice is a square with side length 21, the area becomes a constant value of 21 * 21 = 441 square units for all values of y.
Thus, the volume of the solid is:
V = ∫[0,7] 441 dy = 441 * ∫[0,7] dy = 441 * [y] evaluated from 0 to 7 = 441 * (7 - 0) = 441 * 7 = 3087 cubic units.
Therefore, the volume of the solid is 3087 cubic units.