Problem Solving

What is the approximate minimum length of a rope required to enclose an area of 154 square meters?
A. 154
B. 60
C. 57
D. 50
E. 44

What is the approximate minimum length of a rope required to enclose an area of 154 square meters?
A. 154
B. 60
C. 57
D. 50
E. 4

For any perimeter, the greatest possible area will be yielded if the perimeter forms a CIRCLE.

We can PLUG IN THE ANSWERS, which represent the smallest approximate perimeter required to yield an area of 154.
Since the question stem asks for the smallest possible perimeter, start with the smallest answer choice.
When the correct answer choice is plugged in, the perimeter will form a circle with an approximate area of 154.

E: 44
Here, 2πr = 44, implying that r = 44/(2π) = 22/π ≈ 7.
If this perimeter form a circle, we get:
A = πr² = π(7²) ≈ (22/7)(7²) = 154.
Success!

The correct answer is E.

Variation of the problem above:

What is the approximate minimum length of a rope required to enclose a rectangular area of 154 square meters?
A. 154
B. 60
C. 57
D. 50
E. 44

RULE:
If rectangle R has a perimeter of x units, then the greatest possible area will be yielded if R is a SQUARE with a side of length x/4.

Example: Let p = 40
If L=10 and W=10, then A = 10*10 = 100.
If L=11 and W=9, then A = 11*9 = 99.
If L=12 and W=8, then A = 12*8 = 96.
As the example above illustrates, the greatest possible area is yielded when L=W=10 and R is a SQUARE.

We can PLUG IN THE ANSWERS, which represent the smallest approximate perimeter that will yield an area of 154.
Since the question stem asks for the smallest possible perimeter, start with the smallest answer choice.

E: 44
Here, the greatest possible area will be yielded if this perimeter forms a square with a side of 11.
If s = 11, then A = s² = 11² = 121.
Since the greatest possible area is too small, eliminate E.

D: 50
Here, the greatest possible area will be yielded if this perimeter forms a square with a side of 50/4.
If s = 50/4 = 25/2, then A = s² = (25/2)² = 625/4 ≈ 156.
Success!

The correct answer is D.

[email protected] wrote:If the shape in question is meant to be a polygon, then the answer is D. However, if the area can be a circle, then the answer will change.

Good point.
I've revised my post accordingly.

Dear Rich/ Mitch,
from your both solutions, I observed the word 'minimum' has no effect on the solution.I have seen the one perimeter can give multiple areas, the greatest is when length equals to width. So it does not matter if perimeter is minimum.

Am I right?

Thanks

Mo2men wrote:Dear Rich/ Mitch,
from your both solutions, I observed the word 'minimum' has no effect on the solution.I have seen the one perimeter can give multiple areas, the greatest is when length equals to width. So it does not matter if perimeter is minimum.

Am I right?

Thanks

Let us assume that the rope must form a rectangle.
The usage of the word minimum affects the solution as follows:
Because the prompt asks for the MINIMUM length of rope, the goal is to select the SMALLEST ANSWER CHOICE that can yield an area of 154.
Thus, to ensure that we select the smallest possible answer choice, we must MAXIMIZE the area that can be yielded by each answer choice.
Since the greatest possible area is yielded when L=W, for each answer choice we must calculate the area when L=W and the area of the rectangle is maximized.

GMATGuruNY wrote:

Mo2men wrote:Dear Rich/ Mitch,
from your both solutions, I observed the word 'minimum' has no effect on the solution.I have seen the one perimeter can give multiple areas, the greatest is when length equals to width. So it does not matter if perimeter is minimum.

Am I right?

Thanks

Let us assume that the rope must form a rectangle.
The usage of the word minimum affects the solution as follows:
Because the prompt asks for the MINIMUM length of rope, the goal is to select the SMALLEST ANSWER CHOICE that can yield an area of 154.
Thus, to ensure that we select the smallest possible answer choice, we must MAXIMIZE the area that can be yielded by each answer choice.
Since the greatest possible area is yielded when L=W, for each answer choice we must calculate the area when L=W and the area of the rectangle is maximized.

Dear Mitch,

I have questions:

1- In the above question we find the max area by min the perimeter. Is there a question that asks for max perimeter when area is minimum??

2-In Min/Max questions, I may MAXIMIZE something by MINIMIZING other thing. However, is ALWAYS vice verse true? i.e can I MINIMIZING something by MAXIMIZING other thing. I feel sometimes the logic does not hold true,

3- do we have rule for MIN perimeter and MAX area in triangle questions?

Thanks in advance

Mo2men wrote:Is there a question that asks for max perimeter when area is minimum?

Given a rectangle with a constant area of A, there is no maximum perimeter.
If area = 100, the following options are possible:
L=10, W=10, P = 10+10+10+10 = 40.
L=20, W=5, P = 20+20+5+5 = 50.
L=100, W=1, P = 100+100+1+1 = 202.
L=1000, W=0.1, P = 1000 + 1000 + 0.1 + 0.1 = 2000.2.
L=10000, W=0.01, P = 10000 + 10000 + 0.01 + 0.01 = 20000.02.
And so on.

As illustrated by the examples above, the value of L can be infinitely large, with the result that the perimeter can also be infinitely large.

2-In Min/Max questions, I may MAXIMIZE something by MINIMIZING other thing. However, is ALWAYS vice verse true? i.e can I MINIMIZING something by MAXIMIZING other thing. I feel sometimes the logic does not hold true,

Generally:
To maximize one aspect, we minimize the other aspects.
To minimize one aspect, we maximize the other aspects.

Of course, there could be exceptions.

3- do we have rule for MIN perimeter and MAX area in triangle questions?

Check my posts here:
https://www.beatthegmat.com/perimeter-of ... 79112.html
https://www.beatthegmat.com/largest-poss ... 74809.html

Mo2men wrote:
I have questions:

1- In the above question we find the max area by min the perimeter. Is there a question that asks for max perimeter when area is minimum??

2-In Min/Max questions, I may MAXIMIZE something by MINIMIZING other thing. However, is ALWAYS vice verse true? i.e can I MINIMIZING something by MAXIMIZING other thing. I feel sometimes the logic does not hold true,

3- do we have rule for MIN perimeter and MAX area in triangle questions?

Thanks in advance

I wouldn't worry about these ideas too much: they're a much bigger part of calculus than algebra I, so they'll appear seldom (if ever) on the current GMAT.