Aussie Maths & Science Teachers: Save your time with SmarterEd
The braking distance of a car, in metres, is directly proportional to the square of its speed in km/h, and can be represented by the equation
`text{braking distance}\ = k xx text{(speed)}^2`
where `k` is the constant of variation.
The braking distance for a car travelling at 50 km/h is 20 m.
--- 4 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
a. `k=0.008`
b. `64.8\ text{m}`
a. `text{braking distance}\ = k xx text{(speed)}^2`
| `20` | `=k xx 50^2` | |
| `k` | `=20/50^2=0.008` |
b. `text{Find}\ d\ text{when speed = 90 km/h:}`
`d=0.008 xx 90^2=64.8\ text{m}`
It is known that the volume of a hailstone \((V)\) is directly proportional to the cube of its radius \((r)\).
A hailstone with a radius of 1.25 cm has a volume of 8.2 cm\(^3\).
--- 6 WORK AREA LINES (style=lined) ---
--- 4 WORK AREA LINES (style=lined) ---
a. \(V=4.1984 \times r^{3}\)
b. \(2.3\ \text{cm (2 d.p.)}\)
a. \(V \propto r^3 \ \Rightarrow \ V=kr^3\)
\(\text{Find} \ k \ \text{given} \ \ V=8.2 \ \ \text{when}\ \ r=1.25:\)
| \(8.2\) | \(=k \times 1.25^3\) |
| \(k\) | \(=\dfrac{8.2}{1.25^3}=4.1984\) |
\(\therefore V=4.1984 \times r^{3}\)
b. \(\text{Find \(r\) when \(\ V=51.1\):}\)
| \(51.1\) | \(=4.1984 \times r^3\) |
| \(r\) | \(=\sqrt[3]{\dfrac{51.1}{4.1984}}=2.3\ \text{cm (1 d.p.)}\) |
Energy \((E)\) stored in a spring, measured in joules, varies directly with the square of its compression distance \(d\), measured in centimetres.
When a spring is compressed by 4 cm, it stores 48 joules of energy.
How much energy is stored when the spring is compressed by 7 cm? (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
\(\text{147 joules}\)
\(E \propto d^2 \ \Rightarrow \ E=k d^2\)
\(\text{Find \(k\) given \(\ E=48 \ \) when \(\ d=4\):}\)
| \(48\) | \(=k \times 4^2\) |
| \(k\) | \(=3\) |
\(\text{Find \(E\) when \(\ d=7\):}\)
\(E=3 \times 7^2=147 \ \text{joules}\)
Jacques is a marine biologist and finds that the mass of a crab is directly proportional to the cube of the diameter of its shell.
If a crab with a shell diameter of 15 cm weighs 680 grams, what will be the diameter of a crab that weighs 1.1 kilograms? Give your answer to 1 decimal place. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`17.6\ text(cm)`
| `M` | `prop d^3` | |
| `M` | `= kd^3` |
`text(When)\ \ M=680, \ d=15`
| `680` | `=k xx 15^3` | |
| `k` | `=0.201481…` |
`text(Find)\ \ d\ \ text(when)\ \ M=1100:`
| `1100` | `=0.20148… xx d^3` | |
| `d` | `=root3(1100/(0.20148…))` | |
| `=17.608…` | ||
| `=17.6\ text{cm (to 1 d.p.)}` |
The stopping distance of a car on a certain road, once the brakes are applied, is directly proportional to the square of the speed of the car when the brakes are first applied.
A car travelling at 70 km/h takes 58.8 metres to stop.
How far does it take to stop if it is travelling at 105 km/h? (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`132.3\ text(metres)`
`text(Let)\ \ d\ text(= stopping distance)`
`d \prop s^2\ \ =>\ \ d = ks^2`
`text(Find)\ k,`
| `58.8` | `= k xx 70^2` |
| `k` | `= 58.8/(70^2)= 0.012` |
`text(Find)\ \ d\ \ text(when)\ \ s = 105:`
| `d` | `= 0.012 xx 105^2` |
| `= 132.3\ text(metres)` |
Fuifui finds that for Giant moray eels, the mass of an eel is directly proportional to the cube of its length.
An eel of this species has a length of 25 cm and a mass of 4350 grams.
What is the expected length of a Giant moray eel with a mass of 6.2 kg? Give your answer to one decimal place. (3 marks)
--- 7 WORK AREA LINES (style=lined) ---
`28.1\ text{cm}`
`text(Mass) prop text(length)^3`
`m = kl^3`
`text(Find)\ k:`
| `4350` | `= k xx 25^3` |
| `k` | `= 4350/25^3` |
| `= 0.2784` |
`text(Find)\ \ l\ \ text(when)\ \ m = 6200:`
| `6200` | `= 0.2784 xx l^3` |
| `l^3` | `= 6200/0.2784` |
| `:. l` | `= 28.13…` |
| `= 28.1\ text{cm (to 1 d.p.)}` |