The base of a solid in the first quadrant of the xy plane is a right triangle bounded by the coordinate axes and the line x+y =4. Cross sections of the solid perpendicular to the base are squares. What is the volume, in cubic units , of the solid?
A) 8
B) 32/3
C) 64/3
D) 128/3
perpendicular to a the base is not good enough.
Let's say the squares are perpendicular to the x-axis. Then the length of each side of the square is just the y-coordinate: 4-x
So, adding up the volumes of all those thin plates of thickness dx, we have
v = ∫[0,4] (4-x)^2 dx = 64/3
the same applies if the squares are perpendicular to the y-axis, due to the symmetry of the region.
To find the volume of the solid, we need to integrate the areas of the square cross sections along the height of the solid.
First, let's find the equation of the line x+y = 4. We can rearrange it to y = 4 - x.
The base of the solid is a right triangle bounded by the coordinate axes and the line x+y = 4. Since the line intersects the x-axis at x = 4 and the y-axis at y = 4, the base is a right triangle with legs of length 4.
Let's set up the integral to find the volume. We will integrate the area of a square cross section from the y-coordinate of the base to the top of the solid.
The y-coordinate of the base is 0, and the top of the solid is where the line intersects the y-axis, which is y = 4.
The side length of the square cross section is determined by the perpendicular distance between the line and the y-axis. We can find this distance by solving the equation of the line for x = 0:
y = 4 - x
y = 4 - 0
y = 4
Therefore, at each height y, the side length of the square cross section is 4 - y.
The integral that represents the volume of the solid is:
∫[0,4] (4-y)^2 dy
Let's evaluate this integral:
∫[0,4] (16 - 8y + y^2) dy
= [16y - 4y^2 + (1/3)y^3] | [0,4]
Substituting the limits of integration:
= (16(4) - 4(4^2) + (1/3)(4^3)) - (0 - 4(0^2) + (1/3)(0^3))
= (64 - 64 + (1/3)(64)) - (0 - 0 + 0)
= (1/3)(64)
= 64/3
Therefore, the volume of the solid is 64/3 cubic units.
The correct answer is C) 64/3.
To find the volume of the solid, we need to find the area of the base and multiply it by the height.
The base of the solid is a right triangle bounded by the coordinate axes and the line x+y=4. Since the line intersects the x and y axes at x=0 and y=0 respectively, the base is a triangle with a height of 4 units.
The equation of the line x+y=4 can be rewritten as y=4-x. So, the base has a length of 4 units along the x-axis.
The area of the base of the solid is given by: 1/2 * base * height = 1/2 * 4 * 4 = 8 square units.
Now, let's consider the cross sections perpendicular to the base. The cross sections are squares. Since the cross sections are squares, their side length is equal to the height of the solid.
So, the volume of the solid is given by: volume = area of base * height = 8 * 4 = 32 cubic units.
Therefore, the volume of the solid is 32 cubic units, which corresponds to option B) 32/3.