Question 3(d) Merit

A product with sec, three readings off a drawn graph (continuity, stationary points, a limit), stationary points of a rational function, a related-rates cone, and an optimisation that has to be proved to a given exact value.

Jamie is pouring flour to form a conical pile. The height of the pile is always the same as the diameter of the base of the cone. Flour is being added at a constant rate of 3 cm³ per second. Volume of a cone = (1/3)π r²h.

At what rate is the height increasing when the pile is 4 cm in height?

The 2024 paper, page 12 ↗ Marking schedule, page 5 ↗

Work it out

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

1. 1 The height always equals the diameter. How can you write V using h alone?

   h = 2r, so r = h/2 and V = (1/3)π(h/2)² h = (π h³)/12.

2. 2 Differentiate with respect to h.

   dV/dh = (π h²)/4

3. 3 Chain the rates together at h = 4.

   dh/dt = (dV/dt) ÷ dV/dh = 3 ÷ (π (4)²)/4 = 3/4π ≈ 0.2387

What the marker wanted

• Achieved A correct expression for dh/dt.
• Merit dh/dt = 0.2387, from a calculus method. Units are not needed, and working in r first, then converting, is accepted too.

In our words, from NZQA’s 2024 marking schedule.

How did it go?

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