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Expert: Robi Bhattacharjee - 6/16/2009


Question
The curved surface area of a cylinder is 24/7 times the total flat surface area(top+bottom).If the biggest possible cone is cut from the cylinder, what is the ratio of the curved surface area of the cone to its flat surface area?


Answer
Hi Shruti,

Let the radius of the cylinder be r

Let the height of the cylinder be h

So using our geometry formulas

we have

2(pi)(r)^2 * 24/7 = 2(pi)(r)(h)

canceling things from both sides

r * 24/7 = h.

Now, the biggest possible cone will have its base on one of the bases of the cylinder, and its vertex on the center of the other base.

So its flat area will have area

(pi)(r)^2.

Its other area will be

(pi)(r)sqrt(r^2 + h^2)

but h = 24/7 r

so

curved area = (pi)(r)sqrt(r^2 + 24^2/7^2*r^2)

  = (pi)(r)(r)(25/7)

So finally, when we take the ratio curved/flat we get

(pi)(r)^2(25/7) / (pi)(r)^2 = ratio

canceling we are left with

25/7.

So the answer is 25/7 or 25:7.


I hope this helps,

Robi

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Robi Bhattacharjee

Expertise

I can answer a variety of questions on mathematics. Questions on trigonometry, calculus(preferably single variable), algebra, geometry, and number theory will be answered. I cannot answer questions on abstract branches of mathematics such as group theory. I also cannot answer questions on statistics. In number theory, I can answer questions on congruences, prime numbers, units, functions, and the riemann-zeta function.

Experience

I have studied advanced math my entire life. I started calculus in sixth grade. I have attended numerous math competitions and I am attending math organizations such as the San-Diego math circle. Also, this year I have been invited to the USAMO which is a prestigious math competition (Every year the USAMO invites 500 students from across the USA to participate in this competition. The top 6 go to represent the USA in the International Math Olympiad).

Organizations
I am in the San Diego Math Circle

Education/Credentials
I am entering high school and have received a perfect score and the STAR test 5 times in a row. I also have gotten recognitions in the AMC 10, AIME, Math Counts, and ARML. Additionally, I have won the San Diego Math Olimpiad twice in a row.

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