The ratio of the length to the width of a golden rectangle is (1 + √(5):2:.The dimensions of a garden form a golden rectangle. The width of the garden is 12 feet. Find the length L of the garden. Round your answer to the nearest foot.
You will be solving for the Length L given the proprotion that has w/L = w/L where the first two numbers come from the golden rectangle
(1 + √(5)/2 = w/L
and they give you a width of 12 so put it in, then solve for L : )
(1 + √(5)/2 = 12/L
L(1 + √(5)= 12(2)
L = 24/(1 + √(5)
L = ...
I'm sort of confused...It's multiple choice though...
A.) 7ft
B.) 24ft
C.) 29 ft
D.) 19 ft
So sorry Tom...
I misread the question...
We need
L/W = L/W
so
(1 + √(5)/2= L/12
so
(12)(1 + √(5)=2L
divide both sides by 2
6(1 + √(5)= L
So yes !!
L = 19.416
which is letter D : )
Ah, the golden ratio, nature's favorite design choice! So, to find the length of the garden, we can use the ratio given. If the width is 12 feet, we can set up the equation (1 + √5):2 = L:12. Now, solving for L, we cross multiply to get (1 + √5) * 12 = 2L. Simplifying this, we have 12 + 12√5 = 2L. Dividing both sides by 2, we get L = (12 + 12√5)/2. Simplifying further, L = 6 + 6√5. Rounding to the nearest foot, we have L ≈ 6 + 6(2.236) ≈ 6 + 13.416 ≈ 19.416. So the length of the garden is approximately 19 feet. But hey, if the garden wants to grow an extra 0.416 feet, who am I to stop it?
To find the length L of the garden, we can use the ratio of the golden rectangle, specifically the ratio of length to width.
The ratio of length to width in a golden rectangle is (1 + √(5)):2. In this case, the width of the garden is given as 12 feet.
We can set up a proportion using the given ratio and width of the garden:
(1 + √(5)):2 = L:12
To solve for L, we can cross multiply:
(1 + √(5)) * 12 = 2 * L
12 + 12√(5) = 2L
Next, we can isolate L by dividing both sides of the equation by 2:
(12 + 12√(5)) / 2 = L
6 + 6√(5) = L
Now, we can calculate the length L by substituting the values:
L = 6 + 6√(5)
To round the answer to the nearest foot, we can round the decimal part of the answer to the nearest whole number.