Which is the area of each faces for the rectangular prism.

[germana2006]

Homework Statement

Considering a rectangular prism bounded by six rectangular faces (x=2, y=0.5,
z=1) and a cylinder (r=0.5, heigh h=2) that cut the rectangular prism in x=1
y=0 and z=0 with a inclination of 45 grads in y and z. Which is the area of
each faces for the rectangular prism. I need not the numerically solutions but
the analytically solution.

Homework Equations

The Attempt at a Solution

The area of each face of a rectangular prism is Ar=a*b but now I must rest the
area of the cylinder. The cylinder area is known as Ac=2*pi*r*h. But this is
not the result. If the rectangular prism would be a cube, it would be very
easy: the area of the rectangle minus the area of the semicircle. But in this
case I think I have to consider that the cylinder cut the rectangular prism
with a inclination, but I am not sure.

 

[HallsofIvy]

Homework Statement

Considering a rectangular prism bounded by six rectangular faces (x=2, y=0.5,
z=1)

That's only 3 faces. I would assume the other 3 are x=0, y= 0, z= 0 but you mention x= 1 below. Are the faces x= 1, y= 0, z= 0, x= 2, y= 0.5, z= 1?

and a cylinder (r=0.5, heigh h=2) that cut the rectangular prism in x=1
y=0 and z=0 with a inclination of 45 grads in y and z.

The height is not relevant if the cylinder goes all the way through the prism. But the precise location is. What line is the axis of the cylinder? "45 grads"= [itex]0.45\pi/2[/itex] radians.

Which is the area of
each faces for the rectangular prism.

The area of the faces is trivial. Do you mean the area of each face inside the cylinder?

I need not the numerically solutions but
the analytically solution.

Homework Equations

The Attempt at a Solution

The area of each face of a rectangular prism is Ar=a*b but now I must rest the
area of the cylinder. The cylinder area is known as Ac=2*pi*r*h. But this is
not the result. If the rectangular prism would be a cube, it would be very
easy: the area of the rectangle minus the area of the semicircle. But in this
case I think I have to consider that the cylinder cut the rectangular prism
with a inclination, but I am not sure.

 

[Architeuthis Dux]

The area of each face of the rectangular prism can be calculated by using the formula A = l*w, where l is the length and w is the width of each face. In this case, we have six faces, so we can calculate the area of each face by plugging in the given values for the dimensions.

For the first face (x=2), the length is 2 and the width is 0.5, so the area is A1 = 2*0.5 = 1 square unit.

For the second face (x=1), the length is 1 and the width is 0.5, so the area is A2 = 1*0.5 = 0.5 square units.

For the third face (y=0.5), the length is 2 and the width is 0.5, so the area is A3 = 2*0.5 = 1 square unit.

For the fourth face (y=0), the length is 2 and the width is 1, so the area is A4 = 2*1 = 2 square units.

For the fifth face (z=1), the length is 2 and the width is 0.5, so the area is A5 = 2*0.5 = 1 square unit.

For the sixth face (z=0), the length is 2 and the width is 1, so the area is A6 = 2*1 = 2 square units.

To find the area of the cylinder, we can use the formula Ac = 2*pi*r*h, where r is the radius and h is the height. In this case, the radius is 0.5 and the height is 2, so the area of the cylinder is Ac = 2*pi*0.5*2 = 2*pi square units.

To find the total area of the rectangular prism, we can add up the areas of all six faces and subtract the area of the cylinder since it cuts into the prism. Therefore, the total area of the rectangular prism is A = (A1+A2+A3+A4+A5+A6) - Ac = (1+0.5+1+2+1+2) - 2*pi = 7 - 2*pi square units.

In summary, the area of each face for the rectangular prism is 1, 0.