グローバルビジネスで役立つ数学（16）アルバイトの労働時間（英語版）

Hi, there!

In a scoring hot summer in my first high school grade, I was talked into
daylighting by a friend of mine. It was good pay, but I had to work almost
one entire week under the direct sun.

As the day went by, I got more suntan, and my mother was flabbergasted
by my appearance as if I had become an athletically-inclined student.

“Good going, my son! You’ve got to go out instead of watching TV at home,”
my mother told me every time I got home.

As one of the memories of my high school days, I felt that I became an adult
through the construction work.

Getting the drift above, I want to challenge answering the word problem
related to daylighting. Here comes the question.

Frank Yoshida has kicked off daylighting as a construction worker. He is
dibbling a garden at a rate of \((300 – 6x)\) square feet per hour, where \(x\) is
the number of hours since he started dibbling. If the garden is 900 square
feet, how long will it take him to finish the garden? Round your answer to
the nearest minute.

I want to find how many hours \(H\) it takes Frank to dibble 900 square feet.
In \(H\) hours Frank can dibble

\(\int_{0}^{H}(300 – 6x)dx\) square feet, so I want to find \(H\) such that

\(\int_{0}^{H}(300 – 6x)dx = 900\).
\(900 = \int_{0}^{H}(300 – 6x)dx\)
\(= [300x – 3x^{2}]|^{H}_{0}\)
\(= [300H – 3H^{2}] – [300・0 – 3・(0)^{2}]\)
\(= 300H – 3H^{2}\)

Putting the formula in order, I get
\(3H^{2} – 300H + 900 = 0\)

and dividing through by 3,
\(H^{2} – 100H + 300 = 0\)

Let me use the quadratic formula to solve for \(H\):

\(ax^{2} + bx + c = 0\)
\(x = \frac{-b\pm\sqrt{b^{2} – 4ac}}{2a}\)
\(H^{2} – 100H + 300 = 0\)
\(H = \frac{100\pm\sqrt{(-100)^{2} – 4・1・300}}{2}\)
\(= 50\pm\frac{1}{2}\sqrt{10000 – 1200}\)
\(= 50\pm\frac{1}{2}\sqrt{8800}\)

The two answers are

\(50 + \frac{1}{2}\sqrt{8800}\approx 96.904\)
and
\(50 – \frac{1}{2}\sqrt{8800}\approx 3.096\)

Since 96 hours isn’t realistic amount of time to spend dibbling a garden,
and I keep the answer that’s approximately 3 hours. The amount 0.096
is approximately 0.096✕60 = 5.76 minutes, so rounding to the nearest
minute Frank spent about 3 hours and 6 minutes dibbling the garden.

Am I living a sunny life now?

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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