A cube box is 14 x 12 x 18 cm what is the surface area of the largest size sphere than can fit in this box leave your answer in terms of pi
To find the maximum size sphere that can fit inside the cube box, we need to determine the length of the diagonal of the cube, which will be the diameter of the sphere.
The diagonal of a cube can be found using the Pythagorean theorem. Let's call the length of one side of the cube "s".
Using the Pythagorean theorem, we have:
diagonal^2 = s^2 + s^2 + s^2
diagonal^2 = 3s^2
Since the cube box has dimensions of 14cm x 12cm x 18cm, s = 14cm (or any side length of the cube).
Substituting s = 14cm into the equation, we have:
diagonal^2 = 3(14cm)^2
diagonal^2 = 3(196cm^2)
diagonal^2 = 588cm^2
Taking the square root of both sides to find the diagonal:
diagonal = sqrt(588cm^2)
diagonal ≈ 24.25cm
Therefore, the diameter of the maximum size sphere that can fit inside the cube box is approximately 24.25cm.
The surface area of a sphere can be calculated using the formula: surface area = 4πr^2, where r is the radius of the sphere.
Since the diameter of the sphere is 24.25cm, the radius will be half of this value:
radius = 24.25cm / 2
radius = 12.125cm
Now, we can calculate the surface area of the sphere:
surface area = 4π(12.125cm)^2
surface area = 4π(147.0156cm^2)
surface area ≈ 579.63π cm^2
Therefore, the surface area of the largest size sphere that can fit inside the cube box is approximately 579.63π cm^2.