Can anyone please explain this

[navdeep_bajwa]

MGMAT Challenge Problems
02/13/06

Can anyone please explain this

A right circular cylinder has a radius r and a height h. What is the surface area of the cylinder?

(1) r = 2h - 2lh

(2) h = 15lr - r

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.


the total surface area can be determined by:

SA = Area of circular top + Area of rectangular side + Area of circular bottom

Inserting mathematical expressions for these areas yields

SA = r2 + lw + r2

where r is the radius of the top and bottom circles, l is the length of the rectangle and w is the width. Notice that l (in red) can be expressed as the circumference of the circle and w (dotted line) can be expressed as the height of the cylinder

Now the equation becomes:

SA = r2 + 2rh + r2 = 2r2 + 2rh

Now that we have an expression for the surface area, we can begin to analyze the question. It might be tempting at this point to decide that, in order to calculate the SA, we would need to know the values of r and h explicitly. However, after some manipulation of the equation (factoring out 2r)...

SA = 2r2 + 2rh
SA = 2r(r + h)

...we can see that knowing the value of r(r + h) will be enough to determine the SA. In this case, we DON’T need to know the values of r and h explicitly. Rather, determining the value of a combination of r and h will be sufficient. So, the rephrased question becomes:

What is the value of r(r + h)?

Statement (1), r = 2h - 2lh, cannot be manipulated to isolate r(r +h). We can also see that different values of h we yield different surface areas. For example:
If h = 2, then r = 3 and the surface area is 30.
But if h = 4, then r = 7.5 and the surface area is (172.5).

So statement (1) does not give one and only one value for the surface area. Statement (1) is therefore NOT sufficient.

Statement (2), however can be manipulated as follows:

h = 15lr - r
(multiply through by r)
(can anyone explain how lr*r becomes 1)
hr = 15 - r2
(add r2 to both sides)
r2 + hr = 15
(factor out an r)
r(r + h) = 15

Statement (2) is sufficient, and the correct answer is B.

[andrew]

Hi... i'm not gonna lie this just drove me crazy too... I thought because l=2Ï€r there was a way to break it down there... it turns out the 15lr-r is actually 15/r-r... the fraction/division sign just looks exactly like the 'l' they use in the problem...

so (15/r-r)*r = 15-r²

i'm sure you get it from there....

If that helped please take a look at the post I made yesterday "Consecutive integers, n is an integer"

[navdeep_bajwa]

Thanks a lot