Question

A solid metal cylinder has a radius r and height h. It is melted down and recast into a new cylinder where the radius is reduced by 5

• A. 1:1
• B. 1:2
• C. 2:1
• D. 4:1

Hint:

When computing scaling ratios for a cylinder volume ( r^2 h), if the radius changes by a scale factor of k and height changes by m, the new volume changes by a factor of k^2 m. Here, k = 12 and m = 4. So, the new volume factor is (12)^2 4 = 14 4 = 1. A scale factor of 1 directly implies a 1:1 ratio.

Answer 1

The Correct Option is A

Concept: The volume (V) of a standard geometric cylinder with base radius r and height h is given by the formula: \[ V = \pi r^2 h \] When an object is melted and recast without any material loss, the relationship between dimensions determines the changes in total volume.

Step 1: Find the volume of the original cylinder (V_1). Given that the original cylinder has a radius of r and a height of h, its volume is: \[ V_1 = \pi r^2 h \]

Step 2: Determine the dimensions of the newly recast cylinder.

• The original radius is reduced by 50 \[ r_2 = r - 50 \]

• The original height is increased by 4 times, so the new height (h_2) becomes: \[ h_2 = 4h \]

Step 3: Calculate the volume of the new cylinder (V_2). Using the cylinder volume formula with the new dimensions: \[ V_2 = \pi (r_2)^2 h_2 = \pi \left(\frac{r}{2}\right)^2 (4h) \] \[ V_2 = \pi \left(\frac{r^2}{4}\right) (4h) \] The factor of 4 in the denominator cancels completely with the 4 in the numerator: \[ V_2 = \pi r^2 h \]

Step 4: Compute the required ratio. The ratio of the volume of the original cylinder to the volume of the new cylinder is: \[ \frac{V_1}{V_2} = \frac{\pi r^2 h}{\pi r^2 h} = \frac{1}{1} \implies 1:1 \] Thus, the volume remains completely unchanged, which matches Option (A) exactly.