Problem
of the Week
Problem
B and Solution
Seeking
Parts Unknown...

Problem

Sylvana and Roberto divide a \(40\)
m by \(75\) m rectangular lot to form
two yards, as shown in the diagram below.

A rectangle placed horizontally with the top and bottom sides of the rectangle greater in length than the left and right sides. A diagonal line slants down and to the left from the top side to the bottom side of the rectangle, dividing the rectangle into two trapezoids. The trapezoid on the left is labelled Sylvana’s yard and the trapezoid on the right is labelled Roberto’s yard. The top side of Sylvana’s yard has length \(60\) m, the bottom side has length \(30\) m, and the left side has length \(40\) m. The top side of Roberto’s yard has length \(x\) and the bottom side has length \(y\).

The area of Roberto’s yard is \(40\%\) of the total area of the two
properties.

What are the values of \(x\) and
\(y\), the missing dimensions of
Roberto’s yard?

What is the area of each yard?

Solution

From the two sides of the rectangle of length \(75\) m, we must have \(60\text{ m} + x=75\text{ m}\) and \(30\text{ m}+y=75\text{ m}\). Thus, the
missing dimensions of Roberto’s yard are \(x=75-60=15\) m and \(y=75-30=45\) m.

The total area of the two yards is \(40\text{\,m}\times 75\text{\,m}=3000\)
m\(^2\). The area of each yard can be
found in a variety of ways:

The area of Roberto’s yard is \(40\%\) of the total area. Thus, the area of
Roberto’s yard is \(40\%\) of \(3000\), or \(0.4\times 3000=1200\) m\(^2\), and the area of Sylvana’s yard is
\(3000-1200=1800\) m\(^2\).

Alternatively, since the area of Roberto’s yard is \(40\%\) of the total area, the area of
Sylvana’s yard must be \(100\%-40\%=60\%\) of the total area. Thus,
the area of Sylvana’s yard is \(0.6\times
3000=1800\) m\(^2\), and the
area of Roberto’s yard is \(3000-1800=1200\) m\(^2\).

We can find the area of one of the yards, and subtract that from
the total area to find the area of the other yard. We will show how to
find the area of Sylvana’s yard.

Notice that Sylvana’s yard is shaped like a trapezoid. We can
calculate the area of Sylvana’s yard by dividing the trapezoid into a
\(40\) m by \(30\) m rectangle (shown in red) and a
triangle with a base of \(30\) m and a
height of \(40\) m (shown in blue).

The area of the rectangle is \(40 \times 30
= 1200\) m\(^2\) and the area of
the triangle is \(\frac{40 \times 30}{2} =
600\) m\(^2\). Thus, the total
area of Sylvana’s yard is \(1200 + 600 =
1800\) m\(^2\), and so the total
area of Roberto’s yard is \(3000 - 1800 =
1200\) m\(^2\).