Final answer:
To work out the value of x, we need to use the Pythagorean theorem. In a square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. Using the Pythagorean theorem and the formula for the area of a circle, we can solve for x. We get 4.205 cm.
Step-by-step explanation:
To work out the value of x, we need to use the Pythagorean theorem. In a square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.
So, the diagonal of the square is equal to 2x.
Using the Pythagorean theorem, we can write the equation as: (2x)² = d², where d is the diameter of the circle.
Since the diameter is twice the radius, we can write the equation as: (2x)² = (2r)².
Using the formula for the area of a circle, A = πr², we can substitute A = 56 cm² and solve for x.
First, we find the radius of the circle by taking the square root of the area divided by π:
r = √(56 cm² / π) ≈ 4.219 cm.
Since the radius is half the diameter, the diameter is 2 times the radius:
d = 2 × 4.219 cm
≈ 8.438 cm.
Now we can substitute this value of d back into the equation:
(2x)² = (8.438 cm)².
Simplifying the equation, we get:
4x² = 71.002 cm².
Divide both sides of the equation by 4 to solve for x²:
x² = 17.7505 cm².
Finally, take the square root of x² to find the value of x:
x ≈ √(17.7505 cm²)
≈ 4.205 cm.