Practice Geometry with the exercise "How high up does the beer go?"

Learning Opportunities

This puzzle can be solved using the following concepts. Practice using these concepts and improve your skills.

Geometry

Statement

Goal

===== Summary and your task =====

You will be given 3 measurements (in centimeters) of a drinking-glass, sitting on a bar:
• its bottomRadius
• its topRadius = the radius of the lip (this is always larger than bottomRadius)
• its height, glassHeight

And you will be given a volume of beer beerVol in cubic-centimeters (also known as milliliters) that's been poured into the drinking-glass.

How high up (beerHeight) in the drinking-glass does the beer go ?

===== Geometry background: frustum =====

The above-described drinking-glass (upside down) in geometry is called a frustum.

Wikipedia describes it thus:
"A cone with a region including its apex cut off by a plane is called a truncated cone; if the truncation plane is parallel to the cone's base, it is called a frustum."

The volume of conical frustum is:
V = (1/3) * π * h * (r² + r * R + R²)
...where r and R are the radiuses and h is the height.
But remember the radius of the top of the drinking-glass isn't the radius of the top of the beer.

To better understand frustum geometry, you can experiment with:
https://www.omnicalculator.com/math/truncated-cone-volume
https://rechneronline.de/pi/truncated-cone.php

===== Oh, tricky =====

Tips:
Your first instinct might be to try an Analytic solution, by simply plugging numbers into a formula.
But there are 4 elements, and you'd need to know 3 of them; but you only know 2 ☹️

These elements are:
• volume of the beer beerVol
• bottom radius of the drinking-glass bottomRadius
• height of the beer (unknown)
• radius of the drinking-glass at the spot where the top of the beer is (unknown)

So Analytic solution might be a bit tricky.
Perhaps try an Iterative method instead.

===== Notes, sources =====

• Beer foam doesn't matter. Think of it as water instead if you prefer
• https://en.wikipedia.org/wiki/Cone#
• https://en.wikipedia.org/wiki/Iterative_method

Input

Line 1: 4 floats separated by a space: bottomRadius topRadius glassHeight beerVol

Output

Line 1: A float, beerHeight rounded and displayed to the nearest tenth
Height is measured perpendicular to the bar

Constraints

0 < beerHeight < glassHeight < 50
0 < bottomRadius < topRadius < 50
0 < beerVol < 5000

Example

Input

2.5 4 15 473

Output

14.3

Solve it

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