# Problem of the Week Problem C and Solution Dollhouse

## Problem

The first level of a dollhouse is in the shape of a rectangle. Its floor plan is shown in the following diagram.

Both the mudroom and the kitchen are square with areas of $$400\text{ cm}^2$$ and $$2500\text{ cm}^2$$, respectively. The living room is rectangular with an area of $$3000\text{ cm}^2$$.

Determine the area of the rectangular storage room.

## Solution

Let the width of a room be the vertical length of the room on the diagram. Let the length of a room be the horizontal length of the room on the diagram.

The kitchen is a square and has an area of $$2500\text{ cm}^2$$. Its length and width must both be $$50$$ cm since $$50\times 50=2500\text{ cm}^2$$. The living room and kitchen have the same width. So the width of the living room must also be $$50$$ cm. But the area of the living room is $$3000 \text{ cm}^2$$, so the length of the living room is $$60$$ cm since $$50\times 60=3000\text{ cm}^2$$.

The mudroom is a square and has an area of $$400\text{ cm}^2$$. Its length and width must both be $$20$$ cm since $$20\times 20=400 \text{ cm}^2$$. The mudroom and storage room have the same width. So the width of the storage room must also be $$20$$ cm.

Now the length of the whole house can be calculated in two ways. We will equate these two expressions to find the length of the storage room. \begin{align} \text{mudroom length + storage room length} &= \text{living room length + kitchen length}\\ 20+\text{storage room length} &= 60+50\\ 20+\text{storage room length}&= 110\\ \text{storage room length}&= 90 \text{ cm} \end{align}

Since the width of the storage room is $$20$$ cm and the length of the storage room is $$90$$ cm, the area of the storage room is $$20\times 90=1800\text{ cm}^2$$.