NeoPets Riddles (Lenny Conundrums) and Answers Here

Thanks everybody. Sorry I couldn't be of more help today. I'm not too good at the math ones. That's why I like the word/thinking ones.

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stormygoddess wrote:

I've got side length of the pentagon as 21.5665 cm.
And the angles of a pentagon are all 108... so the angle of the triangle is 82 degrees.

From here.. it's trig.. and I don't know what to do. Anyone else?

OK, so, with base length 21.5665 and angle 82* I got side lengths of 77 and the total area of 822.12 ??

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stormygoddess wrote:

stormygoddess wrote:
I've got side length of the pentagon as 21.5665 cm.
And the angles of a pentagon are all 108... so the angle of the triangle is 82 degrees.

From here.. it's trig.. and I don't know what to do. Anyone else?

OK, so, with base length 21.5665 and angle 82* I got side lengths of 77 and the total area of 822.12 ??

Crap, my bad. The angle is 72, not 82. Okay, so, side lengths are 35, 35 and 21.5665 and the area comes out to 767.6372 ...

Rounding up, that's 768.

NOT 682.

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wow.
I got an answer about half of that.
i checked my math twice, and still got the same thing.

I must be making a wrong assumption.

Can anyone tell me what I did wrong?

1. I used the pentagon formula, A=5tan(54°)/4, to get one side of the pentagon, a.
2. Then, I used the relationship that the sides of a pentagram are proportional to the Golden Ratio to get the other 2 sides of the triangle, b and c.
3. I then applied Heron's formula to get the area of the triangle, based on the 3 sides: a,b,c.

I then went through a slightly different way.
I applied (2) above to determine the side of the triangle which contains the pentagon.

I applied Heron's formula again to the new triangle, subtracted out the area of the pentagon, and then divided that by 2.

I got the exact same answer...
Did I just make a math mistake?

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nooneatall wrote:

well they cheated my friends out of right answer because they has 2 spaces in between the 2 names, so in my eyes neopets is nothing but a pack of cheating stealling dirty worthless pigs

It may be just me, but I find it hard to see how they are cheating when they can't possibly foresee every correct but misspelled answer. If the answer to a question is 42, how many ways can you type that?

"42"
"forty-two"
"Forty Two"
"FORTY - TWO"
"THE ANSWER IS 42 GIVE ME MY AVVIE!!!1"

How can you say they "cheat" when it's a matter of failing to predict that you [or in this case, your friends] typed the answer in a way that the checker couldn't predict? That's why they make it a point to say how they want the answer written, and you should be careful when you enter your response. They even prompt you before submitting your answer to check what you typed! If you know there were extra spaces in your response that could disqualify you, DON'T PUT THEM IN! Confused

Really, you shouldn't worry so much about a game or let little things like that bother you to the point of foaming at the mouth.

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Here's what I got:

The area of an Isosceles triangle is b*[(sqrt(4*a^2 - b^2))/4] where b is equal to the side shared by the pentagon.

So, the answer I got was 358 cm^2.

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zappBrannigan wrote:

wow.
I got an answer about half of that.
i checked my math twice, and still got the same thing.

I must be making a wrong assumption.

Can anyone tell me what I did wrong?

1. I used the pentagon formula, A=5tan(54°)/4, to get one side of the pentagon, a.
2. Then, I used the relationship that the sides of a pentagram are proportional to the Golden Ratio to get the other 2 sides of the triangle, b and c.
3. I then applied Heron's formula to get the area of the triangle, based on the 3 sides: a,b,c.

I then went through a slightly different way.
I applied (2) above to determine the side of the triangle which contains the pentagon.

I applied Heron's formula again to the new triangle, subtracted out the area of the pentagon, and then divided that by 2.

I got the exact same answer...
Did I just make a math mistake?

Zapp, what did you get in step 1?

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answer
i got 346

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Could someone please check this:

Equation (Area of pentagon)
800 = (1/4)a^2 * sqrt[25 + 10 * sqrt(5)]
= (1/4) sqrt25 + 10 * sqrt5 *a^2

Result
a = [8*sqrt2 * 5^ 3/4] / [4*sqrt (5 + 2*sqrt5)]

= 21.5636
base of triangle = 21.5636 (b)

Angle inside pentagon = 108 degrees
thus angle outside = 72 degrees and
vertex of isosceles triangle = 36 degrees (x)

1. b/(2h) = tanx/2
2. h = b/(2tanx/2)
3. A = b^2/(4tanx/2)

Using h = b/(2tanx/2) to find height:
h = 29.6797

put back into 1 for confirmation
b/(2h) = tanx/2
21.5636/(2*29.6797) = 0.363272 correct

thus equation 3

A = [21.5636^2] / [(4* tan 36 degrees) / 2]
= 464.988845 / 1.45308506
= 320.001119

check
area of isoceles triangle = b*h/2
= (21.5636* 29.6797)/2
= (21.5636 * 29.6797) / 2 = 320.000589

My answer was 321

edit: Am not sure I am using correct formulae but all calcs are correct using the above formulae

Interesting sidenote:
Draw line segments from the center of the pentagon to two adjacent vertices. This forms an isosceles triangle where the three angles of the isosceles triangle are 72 degrees, 54 degrees and 54 degrees. (That makes each angle of the pentagon 108 degrees.)

Since the pentagon is made up of five of these triangles, the area of the green triangle in the Lenny C is approximately equal to two-fifths the area of the pentagon. Try it and see!

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Lenny
Well if a few of you could please agree, because I can't do this math to double check for myself, but within the last 2 pages you have given the answers of 822, 768, 358, 346, 321, 681 & 682 .....
Which one if any of them are correct? Please!

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Round 242

The 'Lenny' authors, in Round 242, wrote:

In this diagram, the red part has an area of 800 square centimetres.

What is the area of the green part, in square centimetres? Please round up to the nearest whole number, and submit only the answer.

Many resources, such as Wikipedia, state that the area A of a regular pentagon with side-length "t" is given by:

. . . . .A(pent.) = (t^2 / 4) sqrt[5(5 + 2sqrt[5])]

It may easily be determined (or looked up and confirmed) that the angles inside a regular pentagon measure 108 degrees. The green triangle in the picture is obviously isosceles and, due to its construction (with two of its sides being continuations of sides of the pentagon), its base angles measure 72 degree. Then the angle at the "tip" of the star's points is 180 - 72 - 72 = 180 - 144 = 36 degrees.

Split the green triangle in half by drawing a line from the point's "tip" to the midpoint of the base (being also a side of the pentagon). This forms two right triangles. Given that the length of a side of this pentagon is "t", then the base of one of these right triangles is "t/2". Also, the angle opposite the base measures 36/2 = 18 degrees.

Considering just one of the right triangles, the ratio of the base "t/2" to the height "h" is the tangent of the 18-degree angle. This tangent value may be expressed exactly, according to Wikipedia, as being:

. . . . .tan(18 degrees) = (1/5) sqrt[5(5 - 2sqrt[5])]

Since tan(18 degrees) = (t/2) / h, then:

. . . . .h = (t/2) / tan(18 degrees)

. . . .. . .= (t/2)(5 / sqrt[5(5 - 2sqrt[5])])

The area A of any triangle with base "b" and height "h" is given by:

. . . . .A(rt. tri.) = (1/2) b h

In this case, since the right triangle we've been working with is half of the green triangle, we can multiply the above by "2" to find the area of the green triangle:

. . . . .A(green) = (2/1)(1/2)(t/2)(t/2)(5 / sqrt[5(5 - 2sqrt[5])])

. . . . . . . . . .. . .= (t^2 / 4)(5 / sqrt[5(5 - 2sqrt[5])])

Note that we are given the area of the pentagon as being A(pent.) = 800, so the "area" formula for the pentagon (near the beginning of this post) gives us:

. . . . .800 = (t^2 / 4) sqrt[5(5 + 2sqrt[5])]

. . . . .800 / sqrt[5(5 + 2sqrt[5])] = t^2 / 4

Plugging this into our formula for the area of the green triangle gives us:

. . . . .A(green) = (800 / sqrt[5(5 + 2sqrt[5])])(5 / sqrt[5(5 - 2sqrt[5])])

. . . . . . . . . .. . .= (800 * 5) / (sqrt[5] * sqrt[5 + 2sqrt[5]] * sqrt[5] * sqrt[5 - 2sqrt[5]])

. . . . . . . . . .. . .= (800 * 5) / (5 * sqrt[(5 + 2sqrt[5])(5 - 2sqrt[5])])

. . . . . . . . . .. . .= 800 / sqrt[25 - 4*5]

. . . . . . . . . .. . .= 800 / sqrt[5]

...or about 357.770876.... Rounded up, this would be 358.

Please check my work. Thank you.

Eliz.

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evilness wrote:

...within the last 2 pages you have given the answers of 822, 768, 358, 346, 321, 681 & 682 .....Which one if any of them are correct?

Do the maths yourself, and come to your own conclusion.

Eliz.

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I went about it in a slightly different way, longer and slower and probably more painful, but my answer came out exactly the same as your's stapel. Down to the last decimal.

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i got a answer as 677,7 rounding up to 678 square centimeters. I'm not sure how right it is, so i much control it. Very Happy

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I've also got the same ans as stapel's. I didn't use the method u did though.

I used a mathematical rule called the Golden Ratio, with some trigo involved, since the diagram is a pentagram, or a star pentagon in lame man's term. The Golden Ratio applies here, & I got my ans as 358 square cm (rounded up).

Btw, when we submit the ans, do we need to put in the units, or the numerical no. will do?

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musicologist wrote:

I've also got the same ans as stapel's. I didn't use the method u did though.

I used a mathematical rule called the Golden Ratio, with some trigo involved, since the diagram is a pentagram, or a star pentagon in lame man's term. The Golden Ratio applies here, & I got my ans as 358 square cm (rounded up).

Btw, when we submit the ans, do we need to put in the units, or the numerical no. will do?

ok.. i didn't actually do it, yet both the ways look right to me..the one that got 678 and the one that got 358. anyways, no, you just submit the number.

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now i got 358 and i think that's the right answer.

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the answer is 682.
this is the CORRECT ONE.

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beccaa wrote:

the answer is 682.
this is the CORRECT ONE.

How'd u get to this ans? Mind showing ur solution?

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My Solution.
Formulae involved:

Area of a regular pentagon= [t² x sqrt(25+10 x sqrt(50))] /4, where t is the length of 1 side of the pentagon.

The golden ratio = [1+sqrt(5)]/2 , this only applies to pentagrams, or star pentagons. Find out the rest by googling it up.

Area of triangle = 1/2 x base x perpendicular height

_____________________________________________________________
So with the given information,

800 = [t² x sqrt(25+10 x sqrt(50))] /4
3200 = t² x sqrt[25+10 x sqrt(50)]
t² = 464.9872
t = 21.5636

t is the length of the 1 side of the pentagon. According to the Golden Rule,
t : length of the extended side of the isoceles triangle
= 1 : [1+ sqrt(5)]/2

Therefore the length of extended side of the isoceles triangle
= 21.5636 x [1+ sqrt(5)]/2
= 34.8901

To find the area of the isoceles triangle, we need the perpendicular height to the base. Since we've already found out the length of the base = t = 21.5636, to find the perpendicular height, we can look @ the isoceles triangle as 2 right angled triangles.

According to Pythagoras theorem, a² = b² + c², where a is the hypotenuse, b and c are the lengths of the other 2 sides of the right angled triangle.

So in this case, taking c is the unknown height,
a = 34.8901
b = 21.5636 / 2
= 10.7818

a² = b² + c²
34.8901² = 10.7818² + c²
c² = 1101.1055
c = 33.1829

So the area of the isoceles triangle
= 1/2 x base x height
= 1/2 x t x c
= 1/2 x 21.5636 x 33.1829
= 357.7708
= 358

That's how I got my ans.

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