Question

A shed is shown below. It is 10 feet wide and 20 feet long and has a slanted flat roof (i.e., a plane roof) that is 12 feet high at one corner and 10 feet high at the two corners next to that one. Z (a) [5] Three corners of the roof are given by the points (0, 0, 12), (10, 0, 10), and (0, 20, 10). 12 Find the vector vị from (0, 0, 12) to (10, 0, 10) and The vector v2 from (0, 0, 12) to (0, 20, 10). 10 10 0 Vi= V2= 10 20 Find the cross product vi X V2. Find the equation of the plane that makes up the roof. (b) [1] What is the height of the 4th corner of the roof? (Note that x =10 and y= 20 at that point.) (c) [2] Is the roof a parallelogram? Justify your answer. (d) [3] Use Calculus III methods (a multiple integral) to find the volume of the shed.

Answer 1

(a) The vector v₁ is found by subtracting coordinates of the final point from the starting point i.e 10 i + 0 j - 2 k

Similarly v₂ is given by 0 i + 20 j - 2 k

  v₁ x v₂ = -(20 x -2)i -(10 x -2)j + (10 x 20)k = 40 i + 20 j + 200 k

Equation of a plane is given by a point and the normal to the plane, in the following way:

Let a be a fixed point on the plane, let us take it as (0, 0, 12)

Then (p - a) . n = 0will give us the equation of the plane,

where p is any point and n is the normal vector to the plane.

This is always true because a vector on the plane and the normal vector to the plane are always perpendicular. Also the dot product of perpendicular vectors is always 0.

We take p = (x, y, z) and substitute the normal vector, then the equation simplifies to

(x i + y j + (z-12) k) . (40 i + 20 j + 200 k) = 0

40x + 20y + (z-12) x 200 = 0

Taking out the common factor 20, the equation becomes,

2x + y + 10z = 120

(b) Since we know the equation of the plane, we can get the coordinates of the 4th corner, it has x = 10 and y = 20

Putting these 2 values in our equation we have

20 + 20 +10z = 120

10z = 80

z = 8

i.e height of the 4th corner is 8 feet

(c) For this, we find the vectors of all the 4 sides and if they are 2 pairs of parallel vectors, then the roof is a parallelogram.

We already have v₁ = 10 i + 0 j - 2 k and v₂ = 0 i + 20 j - 2 k

We find the other 2 vectors similarly, the 4th corner has coordinates (10, 20, 8) , coordinates of other 2 adjacent corners are (0, 20 , 10) and (10, 0, 10)

v₃ = 10 i + 0 j - 2 k and v₄ = 0 i + 20 j - 2 k

We see that v₁ and v₃  are parallel vectors and v₂ and v₄ are parallel vectors. Therefore the roof is a parallelogram.