Weird Prelim Max and Min Problem (1 Viewer)

[Marc26]

Source: Cambridge 3 Unit Year 11 - Ex 8E Q16

"A farmer with m dollars to spend is constructing a rectangular paddock PQRS.
The side PQ runs along a river and costs n dollars per metre to fence.
The remaining three sides of the paddock cost r dollars per metre to fence.
Find in terms of m, n and r the lengths of the sides of the paddock in order to maximise its area."


Please show full working if you can be bothered, Thanks in advance

 

[Marc26]

Okay, I got the solutions if anyone was curious.

Cost = ny + (2x+y)r = m
ny + 2xr + ry = m
2xr = m - ny - ry
x = (1/2r)(m - ny - ry)

Area = xy
= (1/2r)(m - ny - ry) x y
= my/2r - (n+t)y^2
A = -(n+r)(y^2 - (my)/2r(m+r) + (m^2)/(16r^2(n+r)^2) + (m^2)(16r^2(n+r))
= -(n+r)(y - (m)/(4r(n+r))^2 + (m^2)/(16r^2(n+r))

Therefore, max area = (m^2)/(16r^2(n+r))
when y = m/(4r(n+r))
Therefore, x = m/4r

Sorry for the long complicated form its in, I don't know how to get Latex or whatever its called. :/