Cardioid in coffee cup (part A)

Calculus Level 3

You may notice there is light reflecting in the above coffee cup that looks like a cardioid (heart).

In fact, it is a cardioid, as illustrated below:

Here come a math question:

Let $A$ be the area bounded by the cardioid, and $B$ the area of the "whole" coffee surface (in the first picture).

Find the value of $\displaystyle\left\lfloor \frac{1000A}{B} \right\rfloor.$

Notation: $\lfloor \cdot \rfloor$ denotes the floor function .

The answer is 666.

1 solution

Chan Lye Lee
Oct 27, 2018

Support the blue curve is modeled as $r=2\left(1-\sin\theta\right)$ , then the circle can be modeled as $x^2+(y+1)^2=3^2$ .

So $\displaystyle\frac{A}{B} = \frac{ \frac{1}{2} \int_{0}^{2\pi} \left(2\left(1-\sin\theta\right)\right)^2 d\theta }{\pi \times (3)^2} =\frac{6\pi}{9\pi} = \frac{2}{3}$ . So $\displaystyle\left\lfloor \frac{1000A}{B} \right\rfloor = \boxed{666}$ .

Well，could you please give me a hint of how to prove the shape is indeed a cardioid？I really want to figure out why.

Haosen Chen - 1 year, 11 months ago

Cardioid in coffee cup (part A)

The answer is 666.

1 solution

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