Question 207692
The ratio of the length of a rectangle to its width is the same as that of the diagonal to the length. 
If the width is 2 , how many units are in the length of the diagonal?
:
the hypotenuse (diagonal) = h,
h = {{{sqrt(L^2 + w^2)}}}
 w=2
h = {{{sqrt(L^2 + 2^2)}}}
h = {{{sqrt(L^2 + 4)}}}
:
"the ratio of the length of a rectangle to its width is the same as that of the diagonal to the length.
{{{L/W}}} = {{{h/L}}}
:
Replace W with 2; and h with {{{sqrt(L^2 + 4)}}}
{{{L/2}}} = {{{(sqrt(L^2 + 4))/L}}}
Cross multiply
L*L = {{{2sqrt(L^2 + 4)}}}
:
L^2 = {{{2sqrt(L^2 + 4)}}}
:
Square both sides:
L^4 = 4(L^2 + 4)
:
L^4 = 4L^2 + 16
:
L^4 - 4L^2 - 16 = 0
:
You have to use the quadratic formula to find L^2
a-1; b=-4; c=-16
The positive solution was L^2 = 6.472
:
Use the value to find the diagonal
h = {{{sqrt(6.472 + 2^2)}}}
h = {{{sqrt(10.472)}}}
h = 3.236 units is the diagonal
:
:
Check solution
L = {{{sqrt(6.472)}}} = 2.544 
{{{L/W}}} = {{{h/L}}}
{{{2.554/2}}} = {{{3.236/2.554}}}
1.272 = 1.272