AS Level Decision mathematics (D1) problem

Question

I've been doing the past papers of my D1 decision mathematics module and I came across a question I couldn't do. The question is part D of the below. I'm not sure how to tackle it as usually you turn the problem into some sort of equality and use that as your constraint, then use the constraints to create your objective function...but how can I get this objective function in the form they want?

(The paper is the D1 paper from January 2010 [$\textit{Don't know exam board - Nethesis}$] for if you want to look at the mark-scheme.)

For clarification, D1 is a module taken by some UK students at ages 16-18 as part of AS levels.

The paper in question

Answer 1

Well you want to maximise the total space you have, and large cupboards ($y$) have 40% more space than standard cupboards ($x$). If you say that for each $x$ you get 1 unit of space, then for each $y$ you get 1.4. Thus to maximise space you maximise $x+1.4y$ because $x+1.4y$ is how many units of space you have. Multiply by 5 and you get $5x+1.4y$.

So: If each $x$ has 5 units of space, then as $y=1.4x$, each $y$ gives 7 units of space. Thus to maximise space I maximise $5x+7y$.

Btw I'm disappointed with those of you who have down voted this. I'll admit D1 isn't exactly the most advanced topic, but that doesn't mean we shouldn't answer a question if we can.

Answer 2

Hint: Say that the volume of a normal cupboard is $a> 0$ (this is a constant). Can you write the total volume obtained with $x$ standard cupboards and $y$ large cupboards as a function of $x,y$ and $a$? (Call that $V(x,y,a)$.) Now, you want to find $x,y$ which maximize $V(x,y,a)$. Observing that $a$ is a constant, can you show (given the expression you got for $V(x,y,a)$) that this is equivalent to maximizing $5x+7y$?