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This is the solution to Mathematical Challenge #001.

“The room is a rectangular prism.”  

Let V=the volume of the room, w=the width of the room, h=the height, l=the length. Since it’s a rectangular prism, V=whl.

“The length of the room is always equal to six metres times the number of apples in the basket in the middle of the room.”

Let a=the number of apples in the basket. So, l=6m*a.

“The height of the room is always equal to the number of apples times feet times e to the power of the width divided by feet.”

So, h=a ft * e^(w/ft).

“The width is equal to the natural log of a randomly generated number x times the number of houses in the neighborhood times the square root of hectacres plus seven inches.”

Let n=the number of houses in the neighborhood. Then, w=ln(x)*n*sqrt(Ha)+7in.

“The number of houses in the neighborhood is ten thousand minus the square of the number of drinks Paul has had.”

Let d=the number of drinks Paul has had. Then, n=10000-d^2.

“Paul drinks twice as many shots as there are apples in the basket.”

So, d=2a.

“The number of apples is kept equal to the width raised to the power of x by a magical genie.”

So, a=w^x.

“Knowing all this, how does the volume change as x changes?”

That is, what is dV/dx? The task at hand is to solve the system of equations generated above. Let’s first gather these equations in one place:

V=whl

l=6m*a

h=a ft * e^(w/ft)

w=ln(x)*n*sqrt(Ha)+7in

n=10000-d^2

d=2a

a=w^x

Since we have each of the dimensions isolated already, we can substitute into the first equation to get:

V=(ln(x)*n*sqrt(Ha)+7in)(a ft * e^(w/ft))(6m*a)

We have n, a, and w isolated, so again we substitute:

V=(ln(x)*(10000-d^2)*sqrt(Ha)+7in)((w^x) ft * e^((ln(x)*n*sqrt(Ha)+7in)/ft))(6m*(w^x))

V=(ln(x)*(10000-(2a)^2)*sqrt(Ha)+7in)(((ln(x)*n*sqrt(Ha)+7in)^x) ft * e^((ln(x)*n*sqrt(Ha)+7in)/ft))(6m*(w^x))

Wait a minute. We have w defined in terms of n, n in terms of d, d in terms of a, and a in terms of w. 

d=2a

d=2(w^x)

d=2(ln(x)*n*sqrt(Ha)+7in)^x

d=2(ln(x)*(10000-d^2)*sqrt(Ha)+7in)^x

While this would be very unpleasant to simplify, we can use dimensional analysis to know that this is an impossible state of affairs. Recall that d is a number. Thus we have a unitless value on the left and some quantity of (sqrt(Ha)+7in)^x on the right. Thus the only set of values satisfying this equation is x=0 and d=2. Thus dV/dx is undefined.