Finding the formulae in terms The cost, $£C$,  of building a circular pond is proportional to the square of its diameter, $d$ meters. A pond with diameter $2$ meters costs $£52$.  
Find the formulae for $C$ in terms of $d$.
Okay so I got $C=d^{52}$ but the mark scheme says $C=d^2$ can someone talk me through why they answer is that?
 A: You need to solve for the constant of proportionality: you are told that $C=kd^2$, and are given an additional piece of information that $52=k \times 2^2$. This tells you that $k=13$, so the answer should be $C=13d^2$
(Sorry but $C=d^{52}$ is certainly not correct!)
A: The problem statement says the cost $C$ is proportional to $d^2$ (the square of the diameter). There must exist a constant $A$ such that $$C = Ad^2$$ This is simply all it means for two things to be directly proportional. Raising $d$ to the power of the cost does not make any sense if you think about it. What would this accomplish? You are given that the value $d=2$ corresponds to a cost $C$ of $52$. If you plug this into the equation you simply get $$52 = A\cdot 2^2$$ This is enough information to solve for $A$, and hence you'll have all the information you need to calculate the cost of a pond given any value of $d$.
A: "The cost, $C$, of building a circular pond is proportional to [something]" means $C = \alpha$[something] for some $\alpha \in \mathbb{R}$.
In this case, the [something] is "the square of its diameter, d", i.e. $d^2$.
Thus we have $C = \alpha d^2$ for some $\alpha$.
Next we solve for $\alpha$ using that $52 = \alpha 2^2$,  so $\alpha = 52/4 = 13$.
Thus $C = 13d^2$.
