Answer
$x= -15 + 15\sqrt 5$
Work Step by Step
The given ratio of $\frac{l}{w}$ is $\frac{1 + \sqrt 5}{2}$
$\frac{30}{w}$ = $\frac{1 + \sqrt 5}{2}$
We cross multiply the fractions
(30)(2) = (1 + \sqrt 5)(w)
We divide both sides by $(1 + \sqrt 5)$
$w = \frac{60}{(1 + \sqrt 5)}$
We multiply the fraction by the conjugate which is $(1 - \sqrt 5)$
$w = \frac{60}{(1 + \sqrt 5)} \times \frac{(1 - \sqrt 5)}{(1 - \sqrt 5)}$
Use FOIL to simplify. FOIL: First (Multiply the first variables in the brackets), outside (Multiply the outer variables), Inside (Multiply the inside variables), Last (Multiply the last variables in the brackets).
$w = \frac{60 - 60 \sqrt 5}{(1 - \sqrt 25)}$
The square root of 25 is 5 because 5 x 5 is 25
$w = \frac{60 - 60 \sqrt 5}{(1 - 5)}$
$w = \frac{60 - 60 \sqrt 5}{(-4)}$
We divide the numerator coefficients by -4
$w = \frac{60}{-4} - \frac{60}{-4} \sqrt 5$
We simplify the division
$x= -15 + 15\sqrt 5$
