Applications of Parabolas (Ex 7F)

Year 10 Mathematics Core · Cambridge Ch 7, §7F · Mr Wong

Name

Class

Date

Learning intentions.

Set up a quadratic model from a worded problem
Read key features in context: vertex = max/min, intercepts = start/landing
Decide on a sensible range of values for the variable

Key results

The turning point gives the maximum or minimum (greatest height, largest area). The $x$-intercepts are where the quantity is zero (ground level, zero area). Choose a range so distances, times and lengths stay positive.

Part A — Reading a graph

1. A stone's height is $h=-(t-3)^2+9$ ($h$ m, $t$ s).

a time at max height

b maximum height

c time it lands ($h=0$)

2. A ball thrown upward has $h=30t-5t^2$ ($h$ m, $t$ s).

a max height

b when it lands

c suitable range of $t$

Part B — A bridge cable

3. A 6 m suspension bridge has cable $h=(d-3)^2+2$, where $h$ m is the cable height and $d$ m is the distance from the left pillar ($0\le d\le6$).

a minimum cable height

b at what $d$

c height at each pillar

Part C — Forming and sketching a model

4. A 24 m fence is bent into a rectangle of width $x$ m.

a length in terms of $x$

b area $A$ in terms of $x$

c suitable values of $x$

d maximum area & dimensions

Parabolas 7F — continued

Part D (a rocket) · Challenge (river enclosure)

Name

Part D — A rocket

5. A toy rocket's height is $h=40t-5t^2$ ($h$ m, $t$ s).

a When does it reach maximum height?

b What is the maximum height?

c When does it return to the ground?

d Suitable range of $t$

6. A garden bed against a wall uses 16 m of edging for the three open sides (width $x$, length $L$).

a Write $L$ in terms of $x$.

b Write the area $A$ in terms of $x$.

c Find the maximum area.

d Dimensions for maximum area

Challenge — river enclosure

7. A farmer fences a rectangular paddock beside a straight river. No fence is needed along the river. With 60 m of fencing for the other three sides, let the width (perpendicular to the river) be $x$ m and the length be $L$ m.

a Write $L$ in terms of $x$.

b Show $A=60x-2x^2$.

c Write $A$ in turning point form.

d Maximum area & dimensions