Round: 178
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? Please round to the nearest whole number, and please submit the answer as just a number. For example, if the answer is 150 square metres, you would just submit the number "150".
Answer: 334888
This is going to be difficult to do without having the luxury of using math symbols, so please bear with me...
First, we need to come up with formulas for what we know. Say that Sa is the area of the square, Ta is the area of the triangle, and Ca is the area of the circle. So:
Sa = 1.75 * Ta
Sa = 1.5 * Ca
Now, let's look at the individual shapes, and come up with formulas of their perimeter as a function of their area. Starting with the square: Ss is the length of one side of the square, and Sp is the perimeter.
Sa = Ss2
Sp = 4 * Ss
Therefore, Sp = 4 * Sa1/2.
For the triangle: (Tp is the perimeter, Ts is the length of one side)
Ta = Ts2 * 31/2/4
Tp = 3 * Ts
Therefore, Tp = 3 * (Ta * 4/(31/2))1/2
For the circle: (Cp is the perimeter/circumference, and Cr is the radius)
Cp = 2 * pi * Cr
Ca = pi * Cr2
Therefore, Cp = 2 * pi * (Ca/pi)1/2
Now, we know that the total perimeter is Cp + Tp + Sp, which equals 4000m. So, substituting the formulas we've found above, we get:
2 * pi * (Ca/pi)1/2 + 3 * (Ta * 4/(31/2))1/2 + 4 * Sa1/2 = 4000m
Now, substitute (1/1.75) * Sa = Ta and (1/1.5) * Sa = Ca:
2 * pi * ((1/1.5) * Sa/pi)1/2 + 3 * ((1/1.75) * Sa * 4/(31/2))1/2 + 4 * Sa1/2 = 4000m
Solve for Sa:
Sa1/2 * (2 * pi * ((1/1.5)/pi)1/2 + 3 * ((1/1.75) * 4/(31/2))1/2 + 4 ) = 4000
Sa = 40002/(2 * pi * ((1/1.5)/pi)1/2 + 3 * ((1/1.75) * 4/(31/2))1/2 + 4 )
And, if you did your math correctly, you should get Sa = 149631.
Now, if you calculate all three areas, they equal Sa + Sa/1.75 + Sa/1.5 which equals 334888 square metres.
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