The sum of length,breadth and height of a rectangular parallelepiped is 20m and area of the whole surface is 121sq units. . Find the length of its diagonal.
Answers
Answer:
The diagonal is 3√31 m long.
This is approximately 16.7 m.
Step-by-step explanation:
Let x, y, z be the length, breadth and height (in metres).
Our destination
First, think about where we're headed. We want the length d of the diagonal. This is given by
d² = x² + y² + z². ... (*)
The important thing is that this is what we need; we do not need to find x, y and z.
In case this is new to you, let's have a quick look at this.
The diagonal is the hypotenuse of a right angled triangle with one leg z and the other leg e being the diagonal inside the xy face, so
d² = e² + z².
That diagonal e in the xy face makes a right angled triangle with legs x and y, so
e² = x² + y².
So we have used Pythagoras' Theorem two times to arrive at the result (*).
Finding x² + y² + z²
The sum of the length, breadth and height is x+y+z = 20.
The surface area is 2(xy+yz+zx) = 121.
Now
( x + y + z )² = x² + y² + z² + 2 ( xy + yz + zx )
=> 20² = d² + 121
=> d² = 400 - 121 = 279 = 9 × 31
=> d = √279 = 3√31
So the diagonal is 3√31 m long.
This is approximately 16.7 m.