3 more ques from ETS

[Shipra]

Hello,

Could anyone please help me understand these questions?

1) Ques No.300 in ETS

The inside dimensions of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical cannister is to be placed inside the box so that it stands upright when the closed box rests on one of its 6 faces. Of all such cannisters that could be used, what is the radius, in inches, of the one that has maximum volume?
a)3
b)4
c)5
d)6
e)8

Answer - b

2) Ques 412

If n is a positive integer and n2(n square) is divisible by 72, then the largest positive integer that must divide n is

a)6
b)12
c)24
d)36
e)48

Answer - b

3) Ques 424

If n is a positive integer less than 200 and 14n/60 is an integer, then n has how many different positive prime factors ?
a)2
b)3
c)5
d)6
e)8

Answer - b

Please tell me if there is some trick to solve questions like (2) and (3)

Thanks,
Shipra

[myun]

Here are tricks you can use to solve (2) and (3).

Q3) 14n/60 = 7n/30 = integer. That means that n is a multiple of 30. First, notice that 30 = 2*3*5, which is made of 3 different prime numbers and that 30 is less than 200. There already we have the answer, b, since the question cannot have more than one correct answer choice.
For n<200, all multiples of 30 which are less than 200 can be expressed as multiples of powers of 2, 3, and 5.

Q2) If you think about it, the largest integer that must divide n is the same as the smallest possible n which satisfies the given conditions. That means the answer choices are the candidates for the integer n itself. But of course, you need to find the smallest possible n from the given 5 choices.

Obviously A) 6 is out because 6^2=36 cannot be divided by 72. But B) 12 satisfies this condition, i.e., 12^2=144 is divisible by 72. This is the smallest possible value of n, so it is the answer.

[Shipra]

Thanks Myun,

The explanations really helped. By any chance do u know the 1st one also??

Shipra

[neetu08]

Hi Shipra
As in the question ,It is not mentioned that on what face the rectangle is placed.
so there are 2 possiblities :
1.base is 6" in that case volume of the cylinder would be
pi*3*3*8
2.base is 8" in that case volume would be
pi*4*4*6
As it is mentioned that volume should be maximum
so it is case 2.And radius is 4"

[Shipra]

Hello Neetu,

Thanks for the explanation. Why don't we consider 10 inches as the base?

Shipra

[jeffq]

It's a rectangular box with the dimensions 6 x 8 x 10.

You have a total of six sides:

2 that are 6" x 8"
2 that are 6" x 10"
2 that are 8" x 10"


Each of these three sides can be used for the base of the box. The problem states that the cylinder is standing upright, therefore, the base of the cylinder is constrained by the dimensions of the base of the box. The height of the cylinder is constrained by the height of the box.

So take the first option...

Base = 6" x 8"
The maximum value for the radius of a cylinder is 3" because it is limited by the 6" side of the box. Since the box is 10" tall, the height of the cylinder cannot exceed 10" here.

Now with the Base = 6" x 10"
The maximum value for the radius here is also 3", again because the cylinder is limited by the 6" side of the box. Now the box is 8" tall, so the height of the cylinder cannot exceed 8".

Base = 8" x 10"
The maximum value for the radius here is 4". But since the height of the box is now 6", the height of the cylinder cannot exceed 6".

[Shipra]

Thanks jeffq,

That helps

[neetu08]

thanx Jeffq
i forgot that 10"base case.good job!!!

[tasimota]

these are pretty fun questions guys. Just use a little imagination and they usually come undone in your mind.

[remPeter]

hi there,

i'm still having trouble understanding myun's explanation for #2. can anyone elaborate? thanks!