geometry, art, architecture and other areas.
Given that the numbers 10 and x are in a golden ratio and that x is larger than 10, determine the exact
value of x. Only algebraic solutions will get full marks.
Anonymous
answered
then works if
(a+b) / a = a / b
here a = x and b = 10
so
(x+10) /x = x / 10
10 (x+10) = x^2
10 x + 100 = x^2
x^2 - 10 x -100 = 0
x = [ 10 +/- sqrt(100 + 400) ] / 2
use the + sign because x must be >10
x = [ 10 + sqrt 500 ] / 2 = (1/2) (10 + 10 sqrt 5) = 5 ( 1+sqrt 5)
Anonymous
answered
Step-by-Step Bot
answered
The golden ratio is given by the equation:
x / 10 = (1 + √5) / 2
First, let's simplify the equation by cross-multiplying:
2x = 10 * (1 + √5)
Next, distribute the 10:
2x = 10 + 10√5
Now, let's isolate x by subtracting 10 from both sides:
2x - 10 = 10√5
Then, divide both sides by 2:
x - 5 = 5√5
Finally, add 5 to both sides to solve for x:
x = 5 + 5√5
Therefore, the exact value of x in the golden ratio with the number 10 is:
x = 5 + 5√5
Explain Bot
answered
Mathematically, this can be expressed as:
x / 10 = (x + 10) / x
To solve this equation, we can start by cross-multiplying:
x * x = 10 * (x + 10)
Simplifying further:
x^2 = 10x + 100
Rearranging the equation:
x^2 - 10x - 100 = 0
Now, we have a quadratic equation. To solve it, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -10, and c = -100. Substituting these values into the formula:
x = (-(-10) ± √((-10)^2 - 4 * 1 * -100)) / (2 * 1)
= (10 ± √(100 + 400)) / 2
= (10 ± √500) / 2
To simplify the radical, we can break down 500 into its factors:
x = (10 ± √(100 * 5)) / 2
= (10 ± √100 * √5) / 2
= (10 ± 10√5) / 2
= 10/2 ± 10√5/2
= 5 ± 5√5
Therefore, the two possible values of x in the golden ratio with 10 (where x is larger than 10) are:
x = 5 + 5√5
or
x = 5 - 5√5
