#
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 \(20\times 90=1800\text{
cm}^2\).