A thin piece of wire 40 meters long is cut into two pieces. One piece

Question

A thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. Which of the following represents the total area, in square meters, of the circular and the square regions in terms of r?

A. \(\pi*r^2\)
B. \(\pi*r^2 + 10\)
C. \(\pi*r^2 + \frac{1}{4}*\pi^2*r^2\)
D. \(\pi*r^2 + (40 - 2\pi*r)^2\)
E. \(\pi*r^2 + (10 - \frac{1}{2}\pi*r)^2\)

A.  \pi*r^2

B.  \pi*r^2 + 10

C.  \pi*r^2 + \frac{1}{4}*\pi^2*r^2

D.  \pi*r^2 + (40 - 2\pi*r)^2

E.  \pi*r^2 + (10 - \frac{1}{2}\pi*r)^2

Bunuel · Accepted Answer

A thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. Which of the following represents the total area, in square meters, of the circular and the square regions in terms of r?

A. \(\pi*r^2\)
B. \(\pi*r^2 + 10\)
C. \(\pi*r^2 + \frac{1}{4}*\pi^2*r^2\)
D. \(\pi*r^2 + (40 - 2\pi*r)^2\)
E. \(\pi*r^2 + (10 - \frac{1}{2}\pi*r)^2\)

A.  \pi*r^2 
B.  \pi*r^2 + 10 
C.  \pi*r^2 + \frac{1}{4}*\pi^2*r^2 
D.  \pi*r^2 + (40 - 2\pi*r)^2 
E.  \pi*r^2 + (10 - \frac{1}{2}\pi*r)^2

The area of a circle will be -  \pi{r^2}  and  2\pi{r}  meters of wire will be used;
There will be  40-2\pi{r}  meters of wire left for a square. Side of this square will be  \frac{40-2\pi{r}}{4}=10-\frac{\pi{r}}{2} , hence the area of the square will be  (10-\frac{\pi{r}}{2})^2 .

The total area will be -  \pi{r^2}+(10-\frac{\pi{r}}{2})^2 .

Answer: E.

PareshGmat · Answer

In this problem, its not at all mentioned that "wire 40 meters long is cut into two equal pieces"

We have to take a variable for distribution of 40 meters in square & circle.

Refer diagram below:

Attachment:

sqa.png [ 3.51 KiB | Viewed 196372 times ]

Let the perimeter of the square = d

then circumference of the circle = 40-d

\(2\pi r = 40-d\)

\(d = 40 - 2\pi r\) ..................... (1)

Area of Circle \(= \pi r^2\) .............. (2)

Each side of square \(= \frac{d}{4} = \frac{40 - 2\pi r}{4} = 10 - \frac{\pi r}{2}\) .................... From (1)

Area of square \(= (10 - \frac{\pi r}{2})^2\)

Total Area \(= \pi r^2 + (10 - \frac{\pi r}{2})^2\)

Answer = E

BrentGMATPrepNow · Answer

Another approach is to plug in a value for r and see what the output should be.

Let's say r =  0 . That is, the radius of the circle = 0
This means, we use the ENTIRE 40-meter length of wire to create the square. 
So, the 4 sides of this square will have length 10, which means the area =  100

So, when r =  0 , the total area =  100

We'll now plug r =  0  into the 5 answer choices and see which one yields an output of  100

A) π( 0 ²) =  0  NOPE
B) π( 0 ²) + 10 =  10  NOPE
C) π( 0 ²) + 1/4(π² *  0 ²) =  0  NOPE
D) π( 0 ²) + (40 - 2π 0 )² =  1600  NOPE
E) π( 0 ²) + (10 - 1/2π( 0 ))² =  100  PERFECT!

Answer: E

Cheers,
Brent

ajit257 · Answer

Bunuel...how did u get side of the square as (40-2pi*r)/4 ?

Bunuel · Answer

40-2\pi{r}  meters of wire are left for the square means that  40-2\pi{r}  is the perimeter of the square so the side of it will be  \frac{40-2\pi{r}}{4}=10-\frac{\pi{r}}{2} .

Study1 · Answer

A thin piece of 40 m wire is cut in two. One piece is a circle with radius of r. Other is a square. No wire is left over. Which represents total area of circle and square in terms of r?

From a 40m rope, two equal parts of 20 represent, respectively, the circumference of Circle C and the perimeter of Square D.

1. Express the perimeter of D in terms of the circumference of C.

Perimeter + Circumference  = 40

Perimeter = 40 - C

2. Substitute accepted identities for perimeter and circumference.

4s = 40 - 2(pi)(r)

3.  Express the side of the square in terms of r.

s = 10 - (pi)(r)/2

4. Write area of circle and square in terms of r.

Area(C) + Area(D) 
= (pi)(r)^2 + s^2 
= (pi)(r)^2 +  (10 - (pi)(r)/2)^2

5.  Verify

2(pi)(r) = 20

r = 10/pi 
   = 3.18 (approx. to hundredth)

s   = (10 - (pi)(r)/2) 
    = 10-   /2
    = 5

The actual value of the side of the square derived from stating area of square in terms of r is consistent with its known perimeter.

Posted from my mobile device

KarishmaB · Answer

Very minimal calculations are required if you assume a convenient value for R e.g. R = 7/2 since  \pi = 22/7 
Circumference in this case  = 2\pi*R = 2*(22/7)*(7/2) = 22 
So length of leftover wire = 18
Side of square = 18/4
Area of square =  (9/2)^2

Now put R = 7/2 in the second half of the options you want to check.
i.e. say you want to check option (E)
 (10 - 1/2\pi R)^2 =  ^2 = (9/2)^2

Option (E) is correct since the area of the square is (9/2)^2 when R = 7/2 (as shown above)

ENAFEX · Answer

Tried to solve this using the numbers given but I am not able to proceed. Please help. Just curious to know. Thanks.

Say the length of each piece = 20.

So Circumference =20
perimeter of square = 4*side
So length of side = 20/4 = 5.

How do I proceed further?

Bunuel · Answer

The length of each piece = 20;

Circumference of the circle is  2\pi{r}=20  -->  r=\frac{10}{\pi}  -->  area=\pi{r^2}=\frac{100}{\pi} ;
Perimeter of the square  4*side=20  -->  side=5  -->  area=5^2=25 ;

The total area is  \frac{100}{\pi}+25 . Now, you should substitute the value of  r=\frac{10}{\pi}  in the answer choices and see which one gives  \frac{100}{\pi}+25 . Answer choice E works.

Hope it helps.

gmarchanda · Answer

Bunuel- the perimeter of the circle and the square are the same, so: 2(pi)r=4a (a being one side of the square). 
a= (pi)r/2
area of square in terms of r: a^2= (pi)^2*r^2/4
wont this mean that option c is also correct? what am I missing?

pqhai · Answer

Hello.

The question only says: A thin piece of wire 40 meters long is cut  into two pieces ,  NOT into same length pieces . Thus, your assumption 2(pi)r=4a is wrong.

Hope it's clear.

TeamGMATIFY · Answer

Given  : 40 meters long wire is cut into two pieces. One circle with radius r and the rest is used to form a square
  Required  : Total area of the circle and the square

Since we are given the length of the wire, we should find the perimeters.

Perimeter of the circle =  2*\pi*r 
Hence left over wire after removing the circle =  40 - (2*\pi*r)

We need to form a square from this length. 
In other words, this is the perimeter of the square and each side of a square is equal.

Hence side of square =  \frac{1}{4}(40 - (2*\pi*r))  =  10 - \frac{1}{2}\pi*r

Area of the square =  (10 - \frac{1}{2}\pi*r)^2

Total required area =  \pi*r^2 + (10 - \frac{1}{2}\pi*r)^2 
Option E

ScottTargetTestPrep · Answer

We are given that a thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius r and the other is used to form a square.
 
Since the circumference of a circle with radius r is 2πr, the amount of wire used to form the circle is 2πr. Thus, we have 40 – 2πr left over to form the square. In other words, the perimeter of the square is 40 – 2πr. However, since we need to calculate the total area of the circular and the square regions, we need to determine the side of the square in terms of r. Since the perimeter of the square is 40 – 2πr, the side of the square is:
 
side = (40 – 2πr)/4
 
side = 10 – (1/2)πr
 
Now we can determine the areas of the circle and the square.
 
Area of circle = πr2
 
Area of square = side^2 = (10 – (1/2)πr)^2
 
Thus, the combined area of the circle and square is πr2 + (10 – (1/2)πr)2.
 
Answer: E

PGTLrowanhand · Answer

Hi GMATters,

Check out my video explanation for this question here!

LlUbC3ZIra0

Enjoy!

Rowan

shivamtibrewala · Answer

Could you please share some extra questions which are similar to this?

Bunuel · Answer

Bunuel · Answer

23. Geometry 
     Theory   
 Everything you need to know about GMAT Geometry in one slideshow 
 Common Mistakes in Geometry Questions 
 GMAT Trick: Rules about Figures in GMAT Geometry 
 INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT 
 Useful Articles and Practice Questions to ace GMAT Geometry 
 Why Does the GMAT Test Geometry? 
     Questions   
 Geometry Ability Quiz 
 GEOMETRY: Shaded Region Problems 
 Probability and Geometry Problems 
 The E-GMAT Question Series on Geometry 
 DS Questions 
 PS Questions

24. Coordinate Geometry 
     Theory   
 Math: Coordinate Geometry 
 Beauty of Coordinate geometry 
 Dealing with Tangents on the GMAT 
     Questions   
 DS Questions 
 PS Questions

25. Triangles 
     Theory   
 Math: Triangles 
 Mastering Important Concepts Tested by GMAT in Triangles - I 
 Mastering Important Concepts Tested by GMAT in Triangles - II 
 Mastering Important Concepts Tested by GMAT in Triangles - III 
 Medians, Altitudes and Angle Bisectors in Special Triangles on GMAT 
 What Austin Powers Can Teach You about Similar Triangles 
 Determining the Area of Similar Triangles on the GMAT 
 Looking for Similar Triangles on the GMAT 
 GMAT Tip of the Week: Today's Date in Geometry History - Pythagorean Triplets 
 The Pythagorean Triples Properties You’ll See on the GMAT 
     Questions   
 DS Questions 
 PS Questions

26. Polygons 
     Theory   
 Math: Polygons 
     Questions   
 DS Questions 
 PS Questions

27. Circles 
     Theory   
 Math: Circles 
     Questions   
 DS Questions 
 PS Questions

28. Rectangular Solids and Cylinders 
     Theory   
 Math : 3-D Geometries 
     Questions   
 GEOMETRY: 3-D Geometry Questions 
 DS Questions 
 PS Questions

29. Graphs and Illustrations 
     Theory   
     Questions   
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

CrackverbalGMAT · Answer

Solution:

Area of a circle = πr^2 and

Wire used = Circumference of the wire

=2πr meters

Wire left for a square =40−2πr meters

Side of this square will be 40−2πr/4

=10−(πr/2)

=>Area of the square = (10−πr/2)^2.

The total area = πr^2+(10−πr/2)^2.

(option e)

Devmitra Sen
GMAT SME

Alpha1350 · Answer

Hi,

I was wondering what would happen if they asked the same question but instead of square they used rectangle? 
so perimeter of rectange + Circumference of circle = 40 
2 (l + w ) = 40 - 2 pi r
l + w = (40 - 2 pi r)/2 
Area of circle + area of rec

Pi r^2 + l x w - what would i write under l x w ?

thanks!

A thin piece of wire 40 meters long is cut into two pieces. One piece

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

GMAT Problem Solving (PS) Questions

A thin piece of wire 40 meters long is cut into two pieces. One piece

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia