Help me with this conundrum!!?

Question

Gilbert the Gelert farmer (He's a Gelert who happens to be a farmer, not a farmer who grows Gelerts!) has three fields. One field is an equilateral triangle, one field is a circle, and one field is a square. The square field is 75% larger in area than the triangular field, and 50% larger in area than the circular field. In order to completely fence all three of the fields, exactly 4000 metres of fencing is required.

What is the total area of all three fields, in square metres?

Answer 1

I'm going to call a side of the triangle t and a side of the square s, so here goes. Area of equilateral triangle is A = t^2(sqrt3)/4, perimeter is 3t. Area of a square is A= s^2, perimeter = 4s. Area of a circle is A = pi(r^2), perimeter = 2pi(r).

that last part is telling you that 4000 = 2pi(r) + 4s + 3t

and you have the areas set up as such:

s^2 = (1.75)(t^2 sqrt3 / 4)
and also s^2 = 1.5pi(r^2)

I don't want to do all the work for you, but basically use those last two equations of area to make that first equation for perimeters all in terms of one variable (probably should be s).

Answer 2

The total fencing needed is equivalent to the sum of the perimeters of the fields.

The triangle has side "T", perimeter is 3T
The saquare has side "S", perimeter is 4S
The cirlce has raidus "R", perimeter is 2pi*R

3T + 4S + 2pi*R = 4000 [m]

Square area = S^2
Circle area = pi*R^2
Triangle area = [sqrt(3)/4] * T^2 [*see below]

Square field is 75% larger than triangle field:
(1.75)*[sqrt(3)/4] * T^2 = S^2

Square field is also 50% larger than circle field:
(1.5)*pi*R^2 = S^2

Combine and solve for one variable in terms of another:

(1.75)*[sqrt(3)/4] * T^2 = (1.5)*pi*R^2
T^2 = (4*1.5*pi)/(1.75*sqrt(3)) * R^2
T^2 = 6.22 R^2
T = 2.49 * R

Now solve for "S" in terms of "R"
S^2 = (1.5)*pi*R^2
S = 2.17 * R

Now rewrite the permeter equation, replacing the "S" and "T" with the "R" expressions:

3T + 4S + 2pi*R = 4000 [m]
3(2.49 * R) + 4(2.17 * R) + 2pi*R = 4000 [m]

22.45R = 4000 [m]
R = 178.19 meters

S = 386.82 meters
T = 443.70 meters

So, total area is:
S^2 + pi*R^2 + [sqrt(3)/4] * T^2
=(386.82)^2 + pi*(178.19)^2 + [sqrt(3)/4]*(443.70)^2
=334,627.5 meters^2





*Triangle area = sqrt[m*(m - T)(m - T)(m - T)], (area for any triangle can be found this way)
where m = (1/2)*(T + T + T) = (3T/2)

So, triangle area = sqrt[(3T/2)*(T/2)(T/2)(T/2))]
=sqrt[ (3T^4)/16] = [sqrt(3)/4] * T^2

Answer 3

It's just straight algebra...
The trick is to symbolize the triangle height as base * sin(60)

Here are the results...
The Square area = 149630.615 m2
The Triangle area= 85503.209 m2
The Circle area = 99753.744 m2
Total area ...... 334887.568 m2

The Square side= 386.8212 m
The Triangle base = 444.3661 m
The Triangle height = 384.8323 m
note: height = base * sin(60)
The Circle radius = 178.1926 m

Answer 4

Okay -- I wanted to answer this question because I think the conundrum is a fun word......the only problem is -- I'm not that great at math. Sorry. I had a long day today too so it's not a good time to challenge my mind. My beer is good though. LOL

Answer 5

This can be done but is too much work for 2 points.

Answer 6

Sorry. Can't help you with that one. Personally if I was that farmer, I would be out with my tape measure and measure how many feet I needed.

Answer 7

Isn't this the Lenny conundrum in Neopets? Aren't you supposed to not ask for answers and no one should tell you anyway? Isn't this cheating?

Answer 8

total area = 386.8211671 m^2
side of square = 334887.5677 m

Answer 9

Technically, this is not a conundrum.

What are you doing, trying to get your math homework done for you?!

Answer 10

20000