A room is approximately shaped like a golden rectangle. Its length is 21 ft. What is the room's width? Write your answer in simplified radical form and rounded to the nearest tenth of a foot. Note that the ratio of the length to the width of a golden rectangle is left parenthesis 1 plus StartRoot 5 EndRoot right parenthesis : 2.
21/w = (1 + √5)/2
(1 + √5)w = 42
w = 42/(1+√5) = appr 12.978.. = 13.0 rounded to the nearest tenth
notice that 21 is one of the Fibonacci numbers with 13 as the previous one.
To find the width of the room, we can use the ratio given for a golden rectangle: length to width = (1 + √5):2.
Let's solve for the width using this ratio:
(1 + √5) / 2 = 1.618 / 2 = 0.809
To find the width, we divide the length by this ratio:
Width = Length / Ratio = 21 ft / 0.809 ≈ 25.97 ft
Rounding to the nearest tenth of a foot, the width of the room is approximately 26.0 ft.
To find the width of the room, we can use the ratio of the length to the width of a golden rectangle, which is (1 + √5) : 2.
Let's set up an equation using this ratio. Since the length of the room is 21 ft, and the ratio is (1 + √5) : 2, we can write:
21 / x = (1 + √5) / 2
Here, x represents the width of the room. To solve for x, we can cross-multiply:
21 * 2 = x * (1 + √5)
42 = x + x√5
To isolate x, we need to move the x√5 term to the other side of the equation:
x√5 = 42 - x
Now, let's square both sides of the equation to eliminate the √5:
(x√5)² = (42 - x)²
5x² = (42 - x)²
Expand the right side:
5x² = 42² - 2*x*42 + x²
Combine like terms:
5x² = 1764 - 84x + x²
Subtract 5x² from both sides:
0 = 1764 - 84x - 4x²
Rearrange the terms:
4x² + 84x - 1764 = 0
To solve this quadratic equation, we can factor or use the quadratic formula. Since this equation can be factored easily, let's do that:
4(x² + 21x - 441) = 0
Now let's solve for x by setting each factor equal to zero:
x² + 21x - 441 = 0
(x + 49)(x - 9) = 0
So, we have two possible solutions: x + 49 = 0 or x - 9 = 0.
If we solve for x in each equation:
x + 49 = 0:
x = -49
x - 9 = 0:
x = 9
Since the width of a room cannot be negative, we ignore the first solution. Therefore, the width of the room is 9 ft.
To write the answer in simplified radical form, we can see that 9 is a perfect square, so it can be simplified as a whole number without a radical. Thus, the width is 9 ft.