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 integrate the areas of the cross-sections along the y-axis.
First, let's find the equation of the line that passes through the points (-10,5) and (5,5) to determine the length of the base of the squares.
The equation of a line between two points (x1, y1) and (x2, y2) can be found using the slope-intercept form:
y - y1 = m(x - x1)
Here, the points are (-10, 5) and (5, 5), and the equation becomes:
y - 5 = 0(x - (-10))
Simplifying, we have:
y - 5 = 0
y = 5
This means the line through (-10,5) and (5,5) is a horizontal line at y = 5.
Since the cross-sections perpendicular to the y-axis are squares, the length of each side of the cross-sections is equal to the distance between the y-axis and the line y = 5.
The distance between a point (x, y) and the line y = b can be calculated as:
Distance = |y - b|
In our case, the distance between any point on the y-axis (x, 0) and the line y = 5 is:
|0 - 5| = 5
So, the length of each side of the cross-sections is 5.
Now, let's find the length of the base of the solid by finding the distance between the y-axis and the point (5, 5).
The distance between two points (x1, y1) and (x2, y2) can be calculated as:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
In our case, the distance between the y-axis (x = 0) and the point (5, 5) is:
Distance = √[(5 - 0)^2 + (5 - 0)^2]
Distance = √[25 + 25]
Distance = √50
So, the length of the base of the solid is √50.
To find the volume of the solid, we need to integrate the areas of the cross-sections along the y-axis. The volume can be calculated using the formula:
Volume = ∫[A(y)] dy
Since the cross-sections are squares with a side length of 5, the area of each cross-section is 5^2 = 25.
The limits of integration along the y-axis are from y = 0 to y = 5, since the solid extends from the y-axis to the line y = 5.
Therefore, the volume of the solid can be calculated as:
Volume = ∫[0 to 5] (25) dy
= 25 ∫[0 to 5] dy
= 25 [y] [0 to 5]
= 25 [5 - 0]
= 25 (5)
= 125 cubic units.
Hence, the volume of the solid is 125 cubic units.
To find the volume of the solid, we need to calculate the area of each cross-section perpendicular to the y-axis and then integrate over the range of y-values.
Since the cross-sections are squares, we can find their area by squaring the length of one side. Let's find the side length of these squares.
The vertices of the base triangle are (-10,5), (5,5), and the origin. We can see that the base triangle has a height of 5 units, which means the side length of the square cross-sections is also 5 units.
Now, let's integrate over the y-axis to find the total volume.
The limits of integration along the y-axis are given by the range of y-values of the base triangle. We can see that the base triangle has a constant height of 5 units, so the limits of integration along the y-axis are 0 to 5.
The integral we need to solve is the integral of the area of each cross-section, which is just the area of a square:
Volume = ∫[0 to 5] (side length)^2 dy
Since the side length is 5 units, we can simplify the integral:
Volume = ∫[0 to 5] 5^2 dy
= ∫[0 to 5] 25 dy
Now, let's solve this integral:
Volume = 25y | [0 to 5]
= 25(5) - 25(0)
= 125 cubic units
Therefore, the volume of the solid is 125 cubic units.