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The outline of a sign for an ice-cream store is made by

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The outline of a sign for an ice-cream store is made by [#permalink] New post 05 Apr 2008, 21:17
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The outline of a sign for an ice-cream store is made by placing 3/4 of the circumference of a circle with radius 2 feet on top of an isosceles triangle with height 5 feet, as shown above. What is the perimeter, in feet, of the sign?

(A) 3pi + 3 sqrt 3
(B) 3pi + 6 sqrt 3
(C) 3pi + 3 sqrt 33
(D) 4pi + 3 sqrt 3
(E) 4pi + 6 sqrt 3
[Reveal] Spoiler: OA

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Re: 1000PS: Section 7 Question 18 [#permalink] New post 05 Apr 2008, 22:11
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18. The outline of a sign for an ice-cream store is made by placing 3/4 of the circumference of a circle with radius 2 feet on top of an isosceles triangle with height 5 feet, as shown above. What is the perimeter, in feet, of the sign?
(A) 3pi + 3 sqrt 3
(B) 3pi + 6 sqrt 3
(C) 3pi + 3 sqrt 33
(D) 4pi + 3 sqrt 3
(E) 4pi + 6 sqrt 3


The circumference of the ice cream part is 0.75 * 2 * pi *r = 3pi.

Now the triangle portion. While there might be a way to geometrically calculate this, we can easily arrive at the answer by backsolving. We know that the triangle is an isosceles triangle. This means that, as its drawn, the two sides that would be enclosed by the perimeter are equal.

We also know that the height is 5ft and the base is less than 4 ft. We also have only 3 choices, A B and C. We can eliminate A because the sum of the 2 sides are definetely > 10ft. C cannot be because its way too much. Leaving B as the only answer.
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Re: 1000PS: Section 7 Question 18 [#permalink] New post 22 Nov 2010, 23:54
can someone explain this problem in a better way... Bunuel it would be great if you can take the lead.

thanks
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Re: 1000PS: Section 7 Question 18 [#permalink] New post 23 Nov 2010, 02:52
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can someone explain this problem in a better way... Bunuel it would be great if you can take the lead.

thanks


The outline of a sign for an ice-cream store is made by placing 3/4 of the circumference of a circle with radius 2 feet on top of an isosceles triangle with height 5 feet, as shown above. What is the perimeter, in feet, of the sign?
(A) 3pi + 3 sqrt 3
(B) 3pi + 6 sqrt 3
(C) 3pi + 3 sqrt 33
(D) 4pi + 3 sqrt 3
(E) 4pi + 6 sqrt 3
Attachment:
untitled.PNG
untitled.PNG [ 6.49 KiB | Viewed 7128 times ]
As 3/4 of the circumference of a circle is placed on top of a triangle then the perimeter of big arc AB will be 3/4 of a circumference so \(\frac{3}{4}*2\pi{r}=3\pi\), so the answer is either A, B, or C.

Next, as big arc is 3/4 of a circumference then small arc is 1/4 of a circumference or 1/4*360=90 degrees, so \(\angle{AOB}=90\). Now, as \(AO=BO=r=2\) then \(AB=hypotenuse=2\sqrt{2}\) --> \(BD=\frac{AB}{2}=\sqrt{2}\) --> \(AC=BC=\sqrt{BD^2+DC^2}=\sqrt{2+25}=3\sqrt{3}\).

The whole perimeter equals to big arc AB + AC + BC: \(P=3\pi+3\sqrt{3}+3\sqrt{3}=3\pi+6\sqrt{3}\).

Answer: B.
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Re: 1000PS: Section 7 Question 18 [#permalink] New post 23 Nov 2010, 04:19
Nice explaination Bunuel...
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Re: geometry [#permalink] New post 08 Jul 2011, 21:56
ssarkar wrote:
The outline of a sign for an ice-cream store is made by placing 3/4 of the circumference of a circle with radius 2 feet on top of an isosceles triangle with height 5 feet, as shown above. What is the perimeter, in feet, of the sign?
(A) 3pi +3 sq root 3
(B) 3pi +6 sq root 3
(C) 3pi +2 sq root 33
(D) 4pi +3 sq root 3
(E) 4pi +6 sq root 3



Its B .

3/4 * 2pi*2 = 3pi .........................1

now is the tricky part.
let the centre of circle be O. draw two line OA and OB.
the bigger arc is 3/4( 3/4*360) , hence the smaller will 1/4 = 90
hence traingle OAB is a right angled triangle,
with OA = OB = 2= radius

hence AB= 2*sqrt 2
now we know that D is the mid point of A and B
AC = BC
use pytho. theorem
sqrt( BD^2 + DC^2) = sqrt( 2+25) = 3 sqrt 3

now perimeter of triangle = 2*3sqrt 3 = 6sqrt 3............................2
total perimeter
1+2

3pi + 6 sqrt 3

hence B

PS: Apologies for the awkward diagram .. at office :)
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 20 Jun 2013, 05:39
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 20 Jun 2013, 22:50
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I Will give it a try.
I supposed everyone got 3pi part.
Next its isosceles triangle. Meaning both sides if added should be even.
So among options of 6,3 and 2 u can blindly choose 6 since 2 is absurd here.
Hope it helps.
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 09 Oct 2013, 02:18
Nicely explained Bunuel, Thanks
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 09 Oct 2013, 04:26
Hi ,

I found the answer like this the cirumfrence length will be 3pi (3/4*2pi*2) now the total perimeter will 3pi+2*equal side of the isoleces triangle so our answer will be in the form of 3pi+2*equal side of the isoleces which is only B (3pi+2*(3sqrt3))

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Re: 1000PS: Section 7 Question 18 [#permalink] New post 19 Oct 2013, 23:15
Bunuel wrote:
hirendhanak wrote:
can someone explain this problem in a better way... Bunuel it would be great if you can take the lead.

thanks


The outline of a sign for an ice-cream store is made by placing 3/4 of the circumference of a circle with radius 2 feet on top of an isosceles triangle with height 5 feet, as shown above. What is the perimeter, in feet, of the sign?
(A) 3pi + 3 sqrt 3
(B) 3pi + 6 sqrt 3
(C) 3pi + 3 sqrt 33
(D) 4pi + 3 sqrt 3
(E) 4pi + 6 sqrt 3
Attachment:
untitled.PNG
As 3/4 of the circumference of a circle is placed on top of a triangle then the perimeter of big arc AB will be 3/4 of a circumference so \(\frac{3}{4}*2\pi{r}=3\pi\), so the answer is either A, B, or C.

Next, as big arc is 3/4 of a circumference then small arc is 1/4 of a circumference or 1/4*360=90 degrees, so \(\angle{AOB}=90\). Now, as \(AO=BO=r=2\) then \(AB=hypotenuse=2\sqrt{2}\) --> \(BD=\frac{AB}{2}=\sqrt{2}\) --> \(AC=BC=\sqrt{BD^2+DC^2}=\sqrt{2+25}=3\sqrt{3}\).

The whole perimeter equals to big arc AB + AC + BC: \(P=3\pi+3\sqrt{3}+3\sqrt{3}=3\pi+6\sqrt{3}\).

Answer: B.[/quot



hey bunuel,
as per your attached figure CANT we take the height inside the triangle to be a perpendicular? And as per 30 - 60 -90 theorem we see bc equals 10/sq rt 3, which we rationalize and get side bc as 10*sq rt 3. since this is an equilateral triangle we can get perimiter of 2 sides= (20*sq rt 3)/3=approx 6*sq rt 3..

also please explain how you took BD as half of AB.
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Re: 1000PS: Section 7 Question 18 [#permalink] New post 20 Oct 2013, 22:50
Expert's post
Yash12345 wrote:
Bunuel wrote:
hirendhanak wrote:
can someone explain this problem in a better way... Bunuel it would be great if you can take the lead.

thanks


The outline of a sign for an ice-cream store is made by placing 3/4 of the circumference of a circle with radius 2 feet on top of an isosceles triangle with height 5 feet, as shown above. What is the perimeter, in feet, of the sign?
(A) 3pi + 3 sqrt 3
(B) 3pi + 6 sqrt 3
(C) 3pi + 3 sqrt 33
(D) 4pi + 3 sqrt 3
(E) 4pi + 6 sqrt 3
Attachment:
untitled.PNG
As 3/4 of the circumference of a circle is placed on top of a triangle then the perimeter of big arc AB will be 3/4 of a circumference so \(\frac{3}{4}*2\pi{r}=3\pi\), so the answer is either A, B, or C.

Next, as big arc is 3/4 of a circumference then small arc is 1/4 of a circumference or 1/4*360=90 degrees, so \(\angle{AOB}=90\). Now, as \(AO=BO=r=2\) then \(AB=hypotenuse=2\sqrt{2}\) --> \(BD=\frac{AB}{2}=\sqrt{2}\) --> \(AC=BC=\sqrt{BD^2+DC^2}=\sqrt{2+25}=3\sqrt{3}\).

The whole perimeter equals to big arc AB + AC + BC: \(P=3\pi+3\sqrt{3}+3\sqrt{3}=3\pi+6\sqrt{3}\).

Answer: B.[/quot



hey bunuel,
as per your attached figure CANT we take the height inside the triangle to be a perpendicular? And as per 30 - 60 -90 theorem we see bc equals 10/sq rt 3, which we rationalize and get side bc as 10*sq rt 3. since this is an equilateral triangle we can get perimiter of 2 sides= (20*sq rt 3)/3=approx 6*sq rt 3..

also please explain how you took BD as half of AB.


Triangle ABC is isosceles, thus its height to the base and the median to the base are the same, thus D is the midpoint of AB.

I cannot understand the rest of your post.
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 02 Nov 2013, 09:04
Bunuel wrote:
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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I tried extending the sides of the triangle to make a larger one with the base being the diameter of the triangle.
After that, the given triangle and the new one are similar, and the height is 5 and 7, the base is x and 4, and solved from there using
the Pythagorean theory (height (5)^2 + x/2^2) = side of triangle.
but I got the wrong answer.
Can someone look it over and say if I have a mistake in calculation or a wrong hypothesis?
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 03 Nov 2013, 10:32
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Bunuel wrote:
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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I tried extending the sides of the triangle to make a larger one with the base being the diameter of the triangle.
After that, the given triangle and the new one are similar, and the height is 5 and 7, the base is x and 4, and solved from there using
the Pythagorean theory (height (5)^2 + x/2^2) = side of triangle.
but I got the wrong answer.
Can someone look it over and say if I have a mistake in calculation or a wrong hypothesis?


Not clear what you've done there: "the diameter of the triangle" does not make sense, not clear which is "given triangle" and "new one", why they are similar, etc.

Please attach a diagram and write your reasoning so that it's clear.
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 20 Dec 2014, 02:27
Bunuel wrote:
hirendhanak wrote:
can someone explain this problem in a better way... Bunuel it would be great if you can take the lead.

thanks


The outline of a sign for an ice-cream store is made by placing 3/4 of the circumference of a circle with radius 2 feet on top of an isosceles triangle with height 5 feet, as shown above. What is the perimeter, in feet, of the sign?
(A) 3pi + 3 sqrt 3
(B) 3pi + 6 sqrt 3
(C) 3pi + 3 sqrt 33
(D) 4pi + 3 sqrt 3
(E) 4pi + 6 sqrt 3
Attachment:
untitled.PNG
As 3/4 of the circumference of a circle is placed on top of a triangle then the perimeter of big arc AB will be 3/4 of a circumference so \(\frac{3}{4}*2\pi{r}=3\pi\), so the answer is either A, B, or C.

Next, as big arc is 3/4 of a circumference then small arc is 1/4 of a circumference or 1/4*360=90 degrees, so \(\angle{AOB}=90\). Now, as \(AO=BO=r=2\) then \(AB=hypotenuse=2\sqrt{2}\) --> \(BD=\frac{AB}{2}=\sqrt{2}\) --> \(AC=BC=\sqrt{BD^2+DC^2}=\sqrt{2+25}=3\sqrt{3}\).

The whole perimeter equals to big arc AB + AC + BC: \(P=3\pi+3\sqrt{3}+3\sqrt{3}=3\pi+6\sqrt{3}\).

Answer: B.


HI Bunuel,

I have one doubt here. why are we not adding this length also in calculating parameter AB=hypotenuse=2\sqrt{2}.

Could you please provide u r comments on this.

Thanks
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 21 Dec 2014, 11:09
The outline of a sign for an ice-cream store is made by placing 3/4 of the circumference of a circle with radius 2 feet on top of an isosceles triangle with height 5 feet, as shown above. What is the perimeter, in feet, of the sign?

(A) 3pi + 3 sqrt 3
(B) 3pi + 6 sqrt 3
(C) 3pi + 3 sqrt 33
(D) 4pi + 3 sqrt 3
(E) 4pi + 6 sqrt 3

Answer: B
Circle: 2pi*2*3/4 = 3pi
Triangle: 2*sqrt(25+sqrt(4+4)) = 6 sqrt 3
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 22 Dec 2014, 01:50
Expert's post
PathFinder007 wrote:
Bunuel wrote:
hirendhanak wrote:
can someone explain this problem in a better way... Bunuel it would be great if you can take the lead.

thanks


The outline of a sign for an ice-cream store is made by placing 3/4 of the circumference of a circle with radius 2 feet on top of an isosceles triangle with height 5 feet, as shown above. What is the perimeter, in feet, of the sign?
(A) 3pi + 3 sqrt 3
(B) 3pi + 6 sqrt 3
(C) 3pi + 3 sqrt 33
(D) 4pi + 3 sqrt 3
(E) 4pi + 6 sqrt 3
Image
As 3/4 of the circumference of a circle is placed on top of a triangle then the perimeter of big arc AB will be 3/4 of a circumference so \(\frac{3}{4}*2\pi{r}=3\pi\), so the answer is either A, B, or C.

Next, as big arc is 3/4 of a circumference then small arc is 1/4 of a circumference or 1/4*360=90 degrees, so \(\angle{AOB}=90\). Now, as \(AO=BO=r=2\) then \(AB=hypotenuse=2\sqrt{2}\) --> \(BD=\frac{AB}{2}=\sqrt{2}\) --> \(AC=BC=\sqrt{BD^2+DC^2}=\sqrt{2+25}=3\sqrt{3}\).

The whole perimeter equals to [b]big arc AB + AC + BC
: \(P=3\pi+3\sqrt{3}+3\sqrt{3}=3\pi+6\sqrt{3}\).

Answer: B.


HI Bunuel,

I have one doubt here. why are we not adding this length also in calculating parameter AB=hypotenuse=2\sqrt{2}.

Could you please provide u r comments on this.

Thanks


Because AB is not a part of the perimeter of the sign. Perimeter is the distance around a figure, the length of the boundary.
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 23 Dec 2014, 09:44
Hi, I used another method to find the result but it gives me an approximative result that abviously leads to the B answer:

1- for the circle, the result is: 3pi.
2- for the perimeter of the triangle: the triangle is an isosceles one, the heigh divides it into two 30° 60° 90° triangles. So the length of the sides of each one is: x (short side), x\sqrt{3} ; 2x (hypotenuse).

so when I calculate everything I find: 3pi+(20\sqrt{3}/3). 20/3 yields to 6 (approximately).

Why I don't find the exact result when I use this method?! any answers please
Thanks!
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 25 Dec 2014, 09:13
Bunuel wrote:
hirendhanak wrote:
can someone explain this problem in a better way... Bunuel it would be great if you can take the lead.

thanks


The outline of a sign for an ice-cream store is made by placing 3/4 of the circumference of a circle with radius 2 feet on top of an isosceles triangle with height 5 feet, as shown above. What is the perimeter, in feet, of the sign?
(A) 3pi + 3 sqrt 3
(B) 3pi + 6 sqrt 3
(C) 3pi + 3 sqrt 33
(D) 4pi + 3 sqrt 3
(E) 4pi + 6 sqrt 3
Attachment:
untitled.PNG
As 3/4 of the circumference of a circle is placed on top of a triangle then the perimeter of big arc AB will be 3/4 of a circumference so \(\frac{3}{4}*2\pi{r}=3\pi\), so the answer is either A, B, or C.

Next, as big arc is 3/4 of a circumference then small arc is 1/4 of a circumference or 1/4*360=90 degrees, so \(\angle{AOB}=90\). Now, as \(AO=BO=r=2\) then \(AB=hypotenuse=2\sqrt{2}\) --> \(BD=\frac{AB}{2}=\sqrt{2}\) --> \(AC=BC=\sqrt{BD^2+DC^2}=\sqrt{2+25}=3\sqrt{3}\).

The whole perimeter equals to big arc AB + AC + BC: \(P=3\pi+3\sqrt{3}+3\sqrt{3}=3\pi+6\sqrt{3}\).

Answer: B.



Hi,

I was wondering, did we need to use the angle values etc? Couldn't we just draw the radius from the (assumed) center of the circle to the base of the triangle and then use the pythagorean to find the lengths of the sides?

Tha basis of the trianle is anyway in the circle, and the ends of it are touching the circumference of the circle. We can then draw the 2 radii, to the ends of the base of the trianegle and just use the pythagorean to calculate the base. And from there, move on to the rest of the sides. Right...?
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Re: The outline of a sign for an ice-cream store is made by [#permalink] New post 04 Mar 2015, 02:51
Bunuel wrote:
hirendhanak wrote:
can someone explain this problem in a better way... Bunuel it would be great if you can take the lead.

thanks


The outline of a sign for an ice-cream store is made by placing 3/4 of the circumference of a circle with radius 2 feet on top of an isosceles triangle with height 5 feet, as shown above. What is the perimeter, in feet, of the sign?
(A) 3pi + 3 sqrt 3
(B) 3pi + 6 sqrt 3
(C) 3pi + 3 sqrt 33
(D) 4pi + 3 sqrt 3
(E) 4pi + 6 sqrt 3
Attachment:
untitled.PNG
As 3/4 of the circumference of a circle is placed on top of a triangle then the perimeter of big arc AB will be 3/4 of a circumference so \(\frac{3}{4}*2\pi{r}=3\pi\), so the answer is either A, B, or C.

Next, as big arc is 3/4 of a circumference then small arc is 1/4 of a circumference or 1/4*360=90 degrees, so \(\angle{AOB}=90\). Now, as \(AO=BO=r=2\) then \(AB=hypotenuse=2\sqrt{2}\) --> \(BD=\frac{AB}{2}=





quare_root]2[/square_root]\) --> \(AC=BC=\sqrt{BD^2+DC^2}=\sqrt{2+25}=3\sqrt{3}\).

The whole perimeter equals to big arc AB + AC + BC: \(P=3\pi+3\sqrt{3}+3\sqrt{3}=3\pi+6\sqrt{3}\).

Answer: B.






There is another way to estimate the correct answer-

Circumference of 3/4 of the circle= 3/4* 2*pi*r
= 3/4 * 2*pi*2 (r=2)
= 3 Pi


We are left with answer choices A, B and C

The perimeter of the triangle part has to be even as the sides will be equal ( odd + odd and even + even = even).

only B matches this description

6 sq rt 3 equals 18 (even).

Answer is B!!
Re: The outline of a sign for an ice-cream store is made by   [#permalink] 04 Mar 2015, 02:51
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