Question 3(a) Achieved

Opens with Simpson’s Rule on five measured offsets from a garden path, then a root quotient, a product-of-sines area, an exponential-quotient acceleration integrated to a velocity, and a differential equation that has to be factorised before it will separate.

A garden designer measures a section of garden from a straight path: offsets of 10, 13, 16, 15 and 14 metres, taken at 3-metre intervals along the path.

Using these measurements, and Simpson’s Rule, find an approximation of the area of the garden section.

The 2023 paper, page 10 ↗ Marking schedule, page 5 ↗

See it move: Rectangles, and the integral they become

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1. 1 What is the strip width h, and how many strips are there? Simpson’s rule needs an even number.

   h = 3 m. Five offsets make four strips: even, so Simpson’s rule applies.

2. 2 Simpson’s rule: the ends once, the odd-numbered offsets four times, the even-numbered ones in the middle twice, all times h/3.

   A ≈ (h/3)[y₀ + y₄ + 4(y₁ + y₃) + 2y₂]
   = (3/3)[10 + 14 + 4(13 + 15) + 2(16)]

3. 3 Work it out.

   = 1 × [24 + 112 + 32] = 168

What the marker wanted

• Achieved The estimate, 168 m², from Simpson’s rule. Units are not needed.

In our words, from NZQA’s 2023 marking schedule.

Common mistake

How did it go?

Got it moves it further out. Review brings it back tomorrow.