A solid right circular cylinder of radius 4.5 centimetres and height 12 centimetres just fits completely inside another cylinder so that their axes are perpendicular to each other. What is the radius, in centimetres, of the larger cylinder?

Introduction / Context:
This spatial geometry problem involves one cylinder fitting inside another with their axes perpendicular. The smaller cylinder is placed inside the larger one in such a way that it just fits, touching the inner surface of the larger cylinder. The task is to find the radius of the larger cylinder, given the dimensions of the smaller one. This is equivalent to inscribing a rectangle into a circle in two dimensions.

Given Data / Assumptions:

• The inner cylinder has radius r = 4.5 cm.
• The inner cylinder has height h = 12 cm.
• The inner cylinder fits inside a larger cylinder so that their axes are perpendicular.
• The smaller cylinder just fits, meaning it touches the inner wall of the larger cylinder but does not overlap or leave gaps beyond that circle.
• We must find the radius R of the larger cylinder.

Concept / Approach:
When viewing the configuration along the axis of the larger cylinder, we see the cross section of the larger cylinder as a circle of radius R. The smaller cylinder, whose axis is perpendicular, appears as a rectangle inside this circle. The rectangle has one side equal to the height of the smaller cylinder (12 cm) and the other equal to its diameter (2r = 9 cm). For the smaller cylinder to just fit, this rectangle must be exactly inscribed in the circle, so the circle radius R is half the length of the rectangle diagonal. Using the Pythagoras theorem on this rectangle gives us R.

Step-by-Step Solution:
Step 1: The rectangle representing the smaller cylinder in the cross section has dimensions: width = 2r = 2 * 4.5 = 9 cm and height = h = 12 cm. Step 2: The diagonal d of this rectangle is given by Pythagoras theorem: d^2 = 9^2 + 12^2. Step 3: Compute the squares: 9^2 = 81 and 12^2 = 144, so d^2 = 81 + 144 = 225. Step 4: Thus d = √225 = 15 cm. Step 5: This diagonal is the diameter of the inner circle of the larger cylinder, so the radius R of the larger cylinder is R = d / 2 = 15 / 2 = 7.5 cm.

Verification / Alternative check:
We can verify by noting that in any circle of radius R, the largest possible rectangle that can be inscribed has a diagonal equal to the circle diameter 2R. If R = 7.5 cm, then the diameter is 15 cm, matching the computed diagonal of the 9 by 12 rectangle. Therefore, the rectangle representing the smaller cylinder just fits inside the cross sectional circle of the larger cylinder. Any smaller value of R would make the diagonal of the rectangle longer than the diameter of the circle, which is impossible, and any larger R would leave extra space, meaning it would not be a "just fits" situation.

Why Other Options Are Wrong:
Radii 5 cm, 6 cm and 9 cm do not satisfy the relation R = (1 / 2) * √(9^2 + 12^2). If R = 6 cm, diameter is 12 cm, which is less than the rectangle diagonal of 15 cm, so the rectangle would not fit. A radius of 15 cm would give a diameter of 30 cm, which is much larger than necessary and contradicts the phrase "just fits". Hence these options are not consistent with the geometry of the inscribed rectangle.

Common Pitfalls:
A common mistake is to assume that the radius of the larger cylinder equals the larger of the two dimensions (9 or 12) or half of one of them. This ignores the fact that both dimensions of the rectangle must fit within the circle at the same time, which depends on the diagonal, not just a single side. Another pitfall is to incorrectly apply Pythagoras theorem or to forget to divide the diagonal by two to obtain the radius. Visualising the cross section and remembering that the diameter is the rectangle diagonal helps avoid these errors.

Final Answer:
The radius of the larger cylinder is 7.5 cm.