Chapter 9.8, Problem 20PSC

To determine

To calculate:Thethree dimensions of a rectangular solid.

Answer to Problem 20PSC

The three dimensions are 120, 160 and 200.

Explanation of Solution

Given information:

A rectangular solid with sides 3x , 4x and 5x .

Formula used:

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Calculation:

  

The length of diagonal in terms of a , b and c can be calculated by applying Pythagoras Theorem.

In right angle triangle BDH , we get

  $\begin{array}{l}{d}^2=c^2+{\left(\sqrt{a^2+b^2}\right)}^2\\ d^2=a^2+b^2+c^2\\ d=\sqrt{a^2+b^2+c^2}\end{array}$

HB is a diagonal.

  $\begin{array}{l}HB=\sqrt{a^2+b^2+c^2}\\ 200\sqrt{2}=\sqrt{{(3x)}^2+{(4x)}^2+{(5x)}^2}\\ 200\sqrt{2}=\sqrt{9x^2+16x^2+25x^2}\\ 200\sqrt{2}=\sqrt{50x^2}\\ 200\sqrt{2}=5x\sqrt{2}\\ 200=5x\\ x=40\end{array}$

The dimensions are as follows:

  $\begin{array}{l}\therefore 3x=120\\ \therefore 4x=160\\ \therefore 5x=200\end{array}$