グローバルビジネスで役立つ数学（33）ランプの体積を求める（英語版）

Hi, there!

What could be a mental lamp when you are in depression or in for a sleepless night? Don’t resort to the cheapest way, medication, because it isn’t the way to eradicate the root causes.

Internal problems should be grappled from the inside, or to put it more comprehensively, think of how you should put your mindset to be able to feel easy.

Think of what happiness is. In my light, happiness doesn’t lie in what you have but in how you think. In the same manner, when you are depressed or spending a sleepless night, don’t get into a negative spiral, but just go out of home for a nice breeze and have a mindfulness that you are gradually crawling up from the pits.

Don’t passively wait for your mental lamp to be given from the outside; let your mental lamp lurking deep inside vitalize to cast even a feeble light over your shadow.

Today, let me evaluate the volume of a lamp, not mentally but substantially.

The lamps near the front entrance of an athletic center have octagonal cross sections, where at height \(z\), the area is \(A(z) = 3(2 + \sqrt{2})(2 + z)^{2}\) with \(0 ≤ z ≤ 3\). What is the volume of the lamp? (Source: Harvard University Practice Exams [Slightly changed])

Here is my solution to the problem.

\(3(2 + \sqrt{2})\int_{0}^{3}(2 + z)^{2}dz\)
\(=3(2 + \sqrt{2})\int_{0}^{3}4 + 4z + z^{2}\,dz\)
\(=3(2 + \sqrt{2})[4z + 2z^{2} + \frac{z^{3}}{3}]_{0}^{3}\)
\(=3(2 + \sqrt{2})39\)
\(=117(2 + \sqrt{2})\)

Remarks:
1. octagon = a polygon of eight angles and eight sides
2. cross section = a flat plane cutting through a three-dimensional object,
　usually at right angles to the longest axis of the object

You may check your mindfulness, but don’t “lamp” the computer screen too much; otherwise, your eyesight deteriorates.

Stay tuned, and expect to see my next post.

Keep well.

Frank Yoshida

￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣
【グローバルビジネスで役立つ数学】でもっと学習する b＾＾)
【参考図書】『もう一度高校数学』（著者：高橋一雄氏）株式会社日本実業出版社
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