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Re: A sphere is inscribed in a cube with an edge of 10. What is
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05 Feb 2022, 08:51
Keep in mind that the edge of the cube is equal to the diameter of the sphere inscribed in said cube.
In other words, 10 = edge of the cube = diameter of the sphere.
If we draw the diagonal of the cube, we have that:
Diagonal of the cube = sphere diameter + two equal distances (from the edge to the sphere x 2)
Thus we can express:
Cube diagonal = x + sphere diameter + x
Where x is the distance from the sphere to one of the edges.
And x is the shortest distance between an edge and the sphere, which is what the exercise asks for.
Let us remember that the diagonal of a cube joins two opposite edges of said cube.
So we have to:
Diagonal of the cube = x + 10 +x
Now we must establish the value of the diagonal of a cube of edge 10.
The diagonal of a cube can be constructed:
Let us consider an edge of the cube, if we take the base of the cube, a square of side 20, and obtain its diagonal, we will have that the edge of the cube and the diagonal of the base of the cube form an angle of 90 degrees, and if we draw the diagonal of the cube, to form a triangle with the edge and the diagonal of the base, we will have a right triangle, with leg 1 edge of the cube, leg 2 diagonal of the base of the cube (square of side 10) and the diagonal hypotenuse of the cube.
The diagonal of a square of side 10 is 10√2 (applying Pythagoras it is an isosceles right triangle, whose equal sides are 10, half of a square).
So we have a new rectangle:
Leg 1 (cube edge) = 10
Leg 2 (diagonal of the base of the cube) = 10√2
Hypotenuse = diagonal of the cube.
So applying Pythagoras we have:
10exp2 + (10√2)exp2 = (diagonal cube)exp2
Solving we have:
10exp2 + 10exp2 x 2 = (diagonal cube)exp2
3 x 10exp2 = (diagonal cube)exp2
If we apply square root to both sides, we have:
10√3 = diagonal cube
Now if you know the direct relationship of a cube with edge a, where the diagonal of one of its faces is a√2 and the diagonal of the cube is a√3. It allows you to work much faster. I suggest you, learn the above relationship.
Going back to the situation:
cube diagonal = x + cube edge + x
where x is the requested distance.
10√3 = 2x + 10
Then x = (10√3 -10)/2
x = 5√3 -5
x = 5(√3 -1)
Answer D