AS 91579 · 2023 paper · a rational integral, kinematics, two bounded areas
Question 1(b) Achieved
Practises Differential equations
A linear-plus-reciprocal integral, a velocity in km hr⁻¹ integrated to a distance from a point 3 km away, an area between a root and a parabola, a separable equation, and an area bounded by three different curves at once.
An object’s velocity can be modelled by v(t) = sec2 t, where v is in km hr⁻¹ and t is the time in hours from the start of timing. Initially the object was 3 km from a point P.
Find the distance of this object from the point P after π/4 hours.
| v(t) | = | sec2 t km hr⁻¹ |
|---|---|---|
| initial displacement 3 km from P | ||
| t | = | π/4 hours |
Work it out
Try it on paper first. Stuck? Take one hint at a time.
-
1 Distance from P comes from integrating the velocity. What integrates to sec2 t?
tan t, so s = tan t + c. -
2 Use the start: 3 km from P when t = 0.
3 = tan 0 + c = c, so s = tan t + 3. -
3 Put t = π/4 in.
tanπ/4 = 1, so s = 1 + 3 = 4 km.
What the marker wanted
- Achieved The distance, 4 km, with the integration shown.
In our words, from NZQA’s 2023 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.