let x represent the number of apples and y represent the number of oranges.
TOTAL: x + y = 112
y = 112 - x
PEELED:
3x = 4y
3x = 4(112 - x)
3x = 448 - 4x
7x = 448
x = 64
y = 112 - x
= 112 - 64
= 48
a) There are 64 apples and 48 oranges
Peeled apples: = 3(8) = 24
Unpeeled apples: 64 - 24 = 40
Unpeeled - Peeled: 40 - 24 - 16
b) there are 16 more unpeeled apples than peeled apples
Let L, W, F, H, P represent, respectively, the length, width, number of floors, height, and perimeter of the building. Here are some equations that model the given information.
We can rewrite these into a system of equations involving L, W, F.
Let's look first at the relation between the perimeter and height. The two versions of perimeter are equal to each other, so we have (after substituting for H) ...
... 3H = 2(L+W)
... 3(12F) = 2((W +40) +W) . . . . substitute for H and L
... 36F = 4W +80 . . . . . . . . . . . simplify
... 9F = W +20 . . . . . . . . . . . . . divide by 4
Now, we can look at the cost equations.
... 1180000 = 35000F + 10W(W+40) . . . . . substitute for the various cost terms and for L
... 118000 = 3500F +W² +40W . . . . . . . . . divide by 10
The two simplified equations in W and F are ...
You can solve the first for F and substitute for F in the second. Then you have a quadratic equation in W that can be solved by the usual methods. (Likely, the quadratic formula would be the best choice.) Here, we have elected to let a graphing calculator show the two solutions. One solution has negative dimensions, so is clearly infeasible.
There is one (nearly feasible) solution:
... W ≈ 180.788
... F ≈ 22.31
There is no combination of floor dimensions and integral numbers of floors that will match <em>all</em> of the problem requirements.
_____
A solution that comes about as close as possible is ...
