Question

A metal bar is to be divided into two pieces so that one piece is 3 centimeters longer than twice the length of the other. If the sum of the squares of the two lengths is 109 square centimeters, find the two lengths

Answer 1

Answer:

The two lengths are 3.4303 and 9.8606 centimeters

Step-by-step explanation:

The metal bar is divided into two pieces, so we are going to call X the length of the first piece and Y the length of the second piece.

From the phrase: one piece is 3 centimeters longer than twice the length of the other, we can separate and rewrite as:

One piece - is - 3 centimeters - longer than - twice the length of the other

X                   =               3                      +               2         *       Y

So, X=3+2Y is our first equation

From the phrase: the sum of the squares of the two lengths is 109 square centimeters, we can rewrite as:

The sum of the squares of the two lengths - is - 109 square centimeters

               $X^{2} +Y^{2}$                                            =      109

So, X^2+Y^2=109 is our second equation

Replacing the first equation on the second question we get:

$X^{2} +Y^{2} =109\\(3+2Y)^{2} +Y^{2} =109$

$9+(2*3*2Y)+4Y^{2} +Y^{2} -109=0$

$5Y^{2} +12Y-100=0$

Solving this equation we find two solutions:

Y=3.4303 and Y= -5.8303

Since the question is talking about the length there is no sense use Y=-5.8303, then our first length is 3.4303

So replacing this value on the first equation we get:

X= 3 + 2*Y

X= 3 + 2*3.4303

X= 9.8606

Finally the two length are 3.4303 and 9.8606 centimeters