Answer:
volume = 140 cubic units
area = 183.3 square units
Step-by-step explanation:
I am going to assume we are dealing with a pyramid with a rectangular base. The rectangular base has length 7 units, width 5 units, and height 12 units.
Let's do the volume first since it's simpler.
Volume:
volume of pyramid = (1/3) * (area of base) * height
volume = (1/3) * 7 units * 5 units * 12 units
volume = 140 cubic units
Now we deal with the surface area.
The total surface area of a rectangular pyramid is the sum of the lateral area and the area of the base. The area of the base is the area of a rectangle. The lateral area is the sum of the areas of the 4 triangular sides. Each two opposite sides are congruent. For the base, we have the length and width of the rectangular base, so we can find the area easily.
For the 4 triangular sides, we need to find the the height of the triangles. Since each pair of opposite faces is congruent, we have two triangles we need to find the heights for. I'll call the heights "p" and "q". We use the Pythagorean theorem to find the heights of the triangular faces.
a^2 + b^2 = c^2
(3.5)^2 + (12)^2 = p^2
p^2 = 156.25
p = 12.5
(2.5)^2 + (12)^2 = q^2
q^2 = 150.25
q = 12.25765
Area of triangle = (1/2) * base * height
For two of the triangles, the base is 5 and the height is 12.5.
For the other two of the triangles, the base is 7 and the height is 12.25765.
total surface area = area of base + 2(area of triangle with base 5) + 2(area of triangle with base 7)
A = LW + 2(1/2)(base)(height) + 2(1/2)(base)(height)
A = 7(5) + 2(1/2)(5)(12.5) + 2(1/2)(7)(12.25765)
A = 183.3 square units
. Using your values of x and y, you get 
, which is approximately 9.22. This means your shortest distance is √85≈9.22
