D01-35

Question

What is the approximate minimum length of rope required to enclose an area of 154 square meters?

A. 154
B. 60
C. 57
D. 50
E. 44

Bunuel · Accepted Answer

Official Solution:

What is the approximate minimum length of rope required to enclose an area of 154 square meters?

A. 154
B. 60
C. 57
D. 50
E. 44

Since a circle has the smallest possible perimeter for a given area, enclosing a circular area would be the most efficient way to minimize the length of a rope.
 
Therefore, the 154 square meters should correspond to the area of a circle. To determine the length of the rope, we first need to calculate the radius of a circle with an area of 154 square meters, and then compute the circumference of the circle, which will be equal to the length of the rope. The formula for the area of a circle is  area = \pi r^2 = 154 . We can approximate the radius by using an approximation for  \pi . A common approximation for  \pi  is  \frac{22}{7} . Thus, we have  \frac{22}{7} * r^2 = 154 . This equation yields  r^2 \approx 49 , and consequently,  r \approx 7 .
 
The length of the rope will be equal to the circumference of the circle, which can be calculated as  circumference = 2\pi r \approx 2*\frac{22}{7}*7 = 44  meters.

Answer: E

Bunuel · Answer

1. For a triangle, the minimum possible perimeter for a given area is an equilateral triangle.
2. For a quadrilateral, the minimum possible perimeter for a given area is a square. 
3. For any 2-D shape, so if there is no restriction on the shape, the minimum possible perimeter for a given area is a circle.

vards · Answer

Ohk..!Thanks Bunuel..!Could you please tell which shape be chosen when we are given the area and maximum length of the rope is to be found out ?
Thanks again..!

Bunuel · Answer

You don';t need this for the GMAT, but anyway, for a given area, the perimeter is NOT limited.

For example, consider a rectangle with the area of 100. If the width approaches 0, then the length approaches infinity, which means that the perimeter is also approaches infinity.

skcmadduri · Answer

If I assume a rectangle with 11 X 14 then the perimeter would be 50. Where am I going wrong.

Bunuel · Answer

The question asks to find the approximate   minimum   possible length of a rope, which is 44: 44 < 50.

sanjay1810 · Answer

hello  Bunuel

One quick question as there seems to be contradictory information:

For a given area, which figure has the less possible perimeter?
As per the question above, it's circle.
However, as per your own post on all information on triangles, it's equilateral triangle.
I cant add the link to your own post here as I'm a new member and there seems to be some restriction with posting links.

Please explain. If both are applicable in different scenarios, then do share these scenarios.

Thanks.

Bunuel · Answer

For triangles it's equilateral triangle but for any shape it's circle.

eastcoastchamps · Answer

Hi - per MGMAT guide, it's the square that has the largest area? Could someone help reconcile?

kavyathach · Answer

Hi, is there an explanation as to why a circle is taken instead of a square?

Bunuel · Answer

Because a circle has the minimum possible perimeter for a given area. Foe example, if an area of a 2-D figure is say 10 unit^2, then the shape which has the minimum perimeter for that area is a circle. All other shapes with the area of 10 units^2 will have greater perimeter.

AyeeshaJ · Answer

Hey  Bunuel ,

Thanks for this. Do these three shapes hold true the property that for a given perimeter, they have the highest area?

Look forward to your reply!

Bunuel · Answer

Generally, a regular polygon gives the maximum area for a given perimeter.

For example, a triangle with a given perimeter has a maximum area when it's regular, so when it's equilateral. A rectangle, when it's a square and so on.

For any 2-D shape, a circle has the maximum area for a given perimeter.

AnirudhaS · Answer

I think this is a  high-quality  question.

Navz · Answer

My 2 cents:

More the symmetry of the shape, lower the perimeter for any given area.
In simple words , for a polygon, as the number of edges increases, the symmetry increases.
Circle can be thought of as a shape with infinite number of edges. If we keep increasing the number of edges in a regular polygon we will eventually get a circle.
Triangle ->Square-> pentagon-> hexagon-> heptagon-> octagon->nonagon-> decagon->.................-> Circle

PS: rectangle (2 axes of symmetry) is more symmetric than any triangle (1 axes of symmetry), except equilateral triangle (3 axes of symmetry)

theGisforGorilla · Answer

I think this is a  high-quality  question and I agree with explanation. Excellent question. I fell into the trap of looking for a square.

Off-topic question:  
If you start looking for a rectangle with sides a,b; you get the following:

{ab=154
{2(a+b)=min

{b=154/a
{a+b=min

a+154/a = min
(a^2+154)/a = min

What is the formal way for a mathematician to solve  (a^2+154)/a = min  ?

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

bumpbot · Answer

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