Harry draws a scale diagram of the portion of his garden that is covered in lawn - Leaving Cert Mathematics - Question 5 - 2019

To apply the trapezoidal rule, we first identify the height at each surveyed point along the lawn.

For this diagram, we observe the heights of the sections (from left to right):

• Heights: 0, 2, 15, 18, 15, 12, 15, 18, 15

Using the trapezoidal rule, the area can be calculated by the formula:

$A = \frac{1}{2} \times h \times (y_0 + 2y_1 + 2y_2 + 2y_3 + 2y_4 + 2y_5 + 2y_6 + 2y_7 + y_8)$

Where:

• $h$ is the width of each segment multiplied by the scale it represents,
• $y_i$ are the heights at each segment.

So in our case, applying this:

• $h = 3$ m (since 1 cm = 3 m)
• The heights will be: $0, 2, 15, 18, 15, 12, 15, 18, 15$.
• Thus, substituting in:

$A = \frac{1}{2} \times 3 \times (0 + 2 + 2(15) + 2(18) + 2(15) + 2(12) + 2(15) + 2(18) + 15)$

Calculating the above gives:

$A = \frac{3}{2} \times [0 + 2 + 30 + 36 + 30 + 24 + 30 + 36 + 15] = \frac{3}{2} \times 324 = 486 ext{ m}^2$