LEARN · EXPLAIN · REVISE

In this idealised height model, t is hours after observation begins and h is metres. All trigonometric arguments are in radians. Find the first time the maximum height occurs for 0 ≤ t ≤ 14.

Read the idea, work independently, then explain what changed.

TOPIC 01

Trigonometric ratios and function graphs

The model has period 12 hours and a right shift of 2 hours. The shift marks an upward midline crossing, not the first maximum. This is an invented model, not a measurement record.

PREDICT → MOVE → EXPLAIN

See amplitude, period and phase together.

Predict the number of cycles and the horizontal shift. Switch angle units without changing the curve.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Worked example

In this idealised height model, t is hours after observation begins and h is metres. All trigonometric arguments are in radians. Find the first time the maximum height occurs for 0 ≤ t ≤ 14.

h(t)=4+3\sin\left(\frac{\pi}{6}(t-2)\right)

Working and explanation

BUILD THE REASONING

Hint 1

The greatest sine value is 1. Decide which angles produce it.

Hint 2

Set the angle equal to π/2 + 2πn, then find the earliest allowed time.

Worked solution

Read the maximum height from the midline and amplitude.
$h_{\max}=4+3=7$
Solve the peak condition for time.
$\frac{\pi}{6}(t-2)=\frac\pi2+2\pi n\Rightarrow t=5+12n,\quad n\in\mathbb Z$
Check the observation interval: the preceding peak is before t = 0.
$t=-7,\ 5,\ 17,\ldots\Rightarrow t_{\mathrm{first}}=5$

The first maximum is at t = 5 hours, when the height is 7 m.

Checks and common pitfalls: The model has period 12 hours and a right shift of 2 hours. The shift marks an upward midline crossing, not the first maximum. This is an invented model, not a measurement record.

Think first. Reveal a hint when the class is ready.