I'm not sure whether this question is too difficult, or perhaps it is lacking a diagram, but I have been getting no response to this question, something I had not been not expecting.

Question

I'm not sure whether this question is too difficult, or perhaps it is lacking a diagram, but I have been getting no response to this question, something I had not been not expecting.

To any maths experts out there, I would absolutely LOVE your help, because I just can't get my head around this one.

Here it is:

A shaded circle just fits inside a 2m x 3m rectangle. What is the radius in metres, of the largest circle that will also fit inside the rectangle but will not intersect the shaded circle?

If you believe a diagram would help, I could try and upload the diagram onto photobucket and give you the link.

Be my saviour!

Thanks!

Answer 1

First, the first circle that "just fits" inside the rectangle is tangent to only three sides, the two long sides and one short side. Call its radius R, and the centre A.

The second circle will be tangent to the opposite short side, one of the long sides and the first circle. Call the radius of the second circle r, and the centre B.

Join the centres of the two circles, AB, and drop perpendiculars parallel to the sides of the rectangle until they meet. Call this point C.

Thus we have a right triangle ABC, right-angled at C.

We have the following information:
R=(short side)/2=1
AB=R+r
AC=R-r
BC=sqrt(AB²-AC²)

By equating the length of the long side with the sum of R,BC and r, you can solve for r:
R+BC+r = long side = 3

Post if you need a diagram.

Answer 2

Thanks for the reply.

I'm still stumbling over where point C is. "Drop perpendiculats parallel to the sides of the rectangle until they meet".

A diagram would most definately be helpful.

Once again, thank you for your help.

Answer 3

Thank you for your question! I'll be happy to help you find the answer.

To solve this problem, we need to understand the relationship between the shaded circle and the rectangle. Let's break it down into steps:

Step 1: Understand the shaded circle
First, let's find the radius of the shaded circle. We know that the circle just fits inside the 2m x 3m rectangle, which means that the diameter of the circle is equal to the shorter dimension of the rectangle, which is 2m. Since the diameter is twice the radius, the radius of the shaded circle is 2m/2 = 1m.

Step 2: Find the maximum size of the other circle
Next, we want to find the largest circle that can fit inside the rectangle without intersecting the shaded circle. We can think of this as finding the largest circle that can fit inside the 2m x 3m rectangle, minus the space taken up by the shaded circle.

Step 3: Calculate the maximum radius
To find the maximum radius of the other circle, we need to find the maximum diameter. The diameter of the rectangle is given by the longer dimension, which is 3m. However, we need to subtract the diameter of the shaded circle, which is 2m, from the total diameter of the rectangle, to ensure that the two circles do not intersect.

Therefore, the maximum diameter of the other circle is 3m - 2m = 1m. Since the diameter is twice the radius, the maximum radius of the other circle is 1m/2 = 0.5m.

So, the radius of the largest circle that will fit inside the rectangle without intersecting the shaded circle is 0.5m.

I hope this explanation helps! Let me know if you have any further questions.