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A cube with edge 2 cm has spheres, each of radius 1 cm, at each vertex with vertices as the centres of the spheres. What will be the radius (in cm) of the largest sphere that can be centred within the cube and which touches each of the remaining spheres externally in at most one point?
- a)√2 - 1
- b)√3 - 1
- c)1
- d)√5 - 1
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
A cube with edge 2 cm has spheres, each of radius 1 cm, at each vertex...
Let's consider one of the spheres at the vertex of the cube. We draw a line connecting the center of this sphere to the center of the cube.
[asy]
size(100);
import three;
triple A=(0,0,0),B=(0,1,0),C=(1,1,0),D=(1,0,0),E=(0,0,1),F=(0,1,1),G=(1,1,1),H=(1,0,1);
triple P=(0.5,0.5,0),Q=(0.5,0.5,1);
draw(P--Q,dashed);
draw(A--B--C--D--A--E--F--G--H--E--F--B--C--G--H--D,dashed);
draw(A--E--P--B--F--Q--C--G--D--H--cycle);
draw(Q--H--G--C--Q);
draw(P--A,dashed);
draw(P--B,dashed);
draw(P--C,dashed);
draw(P--D,dashed);
draw(Q--E,dashed);
draw(Q--F,dashed);
draw(Q--G,dashed);
draw(Q--H,dashed);
dot((0,0,0));
dot((1,0,0));
dot((0,1,0));
dot((1,1,0));
dot((0,0,1));
dot((1,0,1));
dot((0,1,1));
dot((1,1,1));
dot((0.5,0.5,0));
dot((0.5,0.5,1));
[/asy]
Let's call the center of this sphere $P$ and the center of the cube $Q$. We want to find the radius of the largest sphere that can be centered within the cube and touches each of the remaining spheres externally in at most one point.
To do this, we need to find the distance between $P$ and $Q$.
We can see that $PQ$ is a diagonal of the cube, so we can use the Pythagorean theorem to find its length.
The length of one side of the cube is $2$ cm, so the length of $PQ$ is $\sqrt{2^2 + 2^2 + 2^2} = \sqrt{12} = 2\sqrt{3}$ cm.
Therefore, the radius of the largest sphere that can be centered within the cube and touches each of the remaining spheres externally in at most one point is $\boxed{\sqrt{3}}$ cm.
[asy]
size(100);
import three;
triple A=(0,0,0),B=(0,1,0),C=(1,1,0),D=(1,0,0),E=(0,0,1),F=(0,1,1),G=(1,1,1),H=(1,0,1);
triple P=(0.5,0.5,0),Q=(0.5,0.5,1);
draw(P--Q,dashed);
draw(A--B--C--D--A--E--F--G--H--E--F--B--C--G--H--D,dashed);
draw(A--E--P--B--F--Q--C--G--D--H--cycle);
draw(Q--H--G--C--Q);
draw(P--A,dashed);
draw(P--B,dashed);
draw(P--C,dashed);
draw(P--D,dashed);
draw(Q--E,dashed);
draw(Q--F,dashed);
draw(Q--G,dashed);
draw(Q--H,dashed);
dot((0,0,0));
dot((1,0,0));
dot((0,1,0));
dot((1,1,0));
dot((0,0,1));
dot((1,0,1));
dot((0,1,1));
dot((1,1,1));
dot((0.5,0.5,0));
dot((0.5,0.5,1));
[/asy]
Let's call the center of this sphere $P$ and the center of the cube $Q$. We want to find the radius of the largest sphere that can be centered within the cube and touches each of the remaining spheres externally in at most one point.
To do this, we need to find the distance between $P$ and $Q$.
We can see that $PQ$ is a diagonal of the cube, so we can use the Pythagorean theorem to find its length.
The length of one side of the cube is $2$ cm, so the length of $PQ$ is $\sqrt{2^2 + 2^2 + 2^2} = \sqrt{12} = 2\sqrt{3}$ cm.
Therefore, the radius of the largest sphere that can be centered within the cube and touches each of the remaining spheres externally in at most one point is $\boxed{\sqrt{3}}$ cm.
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A cube with edge 2 cm has spheres, each of radius 1 cm, at each vertex with vertices as the centres of the spheres. What will be the radius (in cm) of the largest sphere that can be centred within the cube and which touches each of the remaining spheres externally in at most one point?a)√2 - 1b)√3 - 1c)1d)√5 - 1Correct answer is option 'B'. Can you explain this answer?
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A cube with edge 2 cm has spheres, each of radius 1 cm, at each vertex with vertices as the centres of the spheres. What will be the radius (in cm) of the largest sphere that can be centred within the cube and which touches each of the remaining spheres externally in at most one point?a)√2 - 1b)√3 - 1c)1d)√5 - 1Correct answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A cube with edge 2 cm has spheres, each of radius 1 cm, at each vertex with vertices as the centres of the spheres. What will be the radius (in cm) of the largest sphere that can be centred within the cube and which touches each of the remaining spheres externally in at most one point?a)√2 - 1b)√3 - 1c)1d)√5 - 1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A cube with edge 2 cm has spheres, each of radius 1 cm, at each vertex with vertices as the centres of the spheres. What will be the radius (in cm) of the largest sphere that can be centred within the cube and which touches each of the remaining spheres externally in at most one point?a)√2 - 1b)√3 - 1c)1d)√5 - 1Correct answer is option 'B'. Can you explain this answer?.
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