Re-Casting Metal Cylinders

Geometry Level 3

There are two right circular metal cylinders. Cylinder 1 1 has radius r 1 = 1 r_1 = 1 and height h 1 = 2 h_1 = 2 . Cylinder 2 2 has radius r 2 = 2 r_2 = 2 and height h 2 = 4 h_2 = 4 .

The two cylinders are melted down, and their combined volume is re-cast into a new cylinder whose radius and height are the following:

r 3 = r 2 + β h 3 = h 2 + β \large{r_3 = r_2 + \beta \\ h_3 = h_2 + \beta}

What is the value of β \beta ?


The answer is 0.0962498.

Steven Chase by Steven Chase , 37, USA

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1 solution

X X
Jul 28, 2018

Cylinder 1= 1 × 1 × 2 × π = 2 π 1\times1\times2\times\pi=2\pi ,Cylinder 2= 2 × 2 × 4 × π = 16 π 2\times2\times4\times\pi=16\pi ,so the combined volume is ( 2 + β ) 2 ( 4 + β ) π = 18 π (2+\beta)^2(4+\beta)\pi=18\pi .

Solving the equation gets x = 1 3 ( − 8 + 235 − 9 681 3 + 235 + 9 681 3 ) ≈ 0.09625 x=\frac13(-8+\sqrt[3]{235-9\sqrt{681}}+\sqrt[3]{235+9\sqrt{681}})\approx0.09625

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