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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.

A large rectangle with its longer sides placed horizontally.
A horizontal line that is close to the bottom side of the rectangle
divides the rectangle into a top piece and a bottom piece. A vertical
line near the centre of the rectangle divides the top piece into two
rooms: the Living Room on the left and the Kitchen on the right. The
Living Room is larger than the Kitchen. A vertical line close to the
left side of the rectangle divides the bottom piece into two rooms: the
Mudroom on the left and the Storage Room on the right. The Mudroom is
much smaller than the Storage Room.

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