Aussie Maths & Science Teachers: Save your time with SmarterEd
The brightness of a lamp \((L)\) is measured in lumens and varies directly with the square of the voltage \((V)\) applied, which is measured in volts.
When the lamp runs at 7 volts, it produces 735 lumens.
What voltage is required for the lamp to produce 1820 lumens? Give your answer correct to one decimal place. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`11.2\ \text(volts)`
`L prop V^2\ \ => \ \ L=kV^2`
`text(Find)\ k\ \text{given}\ L = 735\ \text{when}\ V = 7:`
| `735` | `= k xx 7^2` |
| `:. k` | `= 735/49=15` |
`text(Find)\ V\ text(when)\ L = 1820:`
| `1820` | `= 15 xx V^2` |
| `V^2` | `= 1820/15=121.33…` |
| `V` | `= sqrt{121.33} = 11.2\ text(volts)\ \ text{(to 1 d.p.)}` |
Moses finds that for a Froghead eel, its mass is directly proportional to the square of its length.
An eel of this species has a length of 72 cm and a mass of 8250 grams.
What is the expected length of a Froghead eel with a mass of 10.2 kg? Give your answer to one decimal place. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`80.1\ text{cm}`
`text(Mass) prop text(length)^2`
`m = kl^2`
`text(Find)\ k:`
| `8250` | `= k xx 72^2` |
| `k` | `= 8250/72^2` |
| `= 1.591…` |
`text(When)\ \ l\ \ text(when)\ \ m = 10\ 200:`
| `10\ 200` | `= 1.591… xx l^2` |
| `l^2` | `= (10\ 200)/(1.591…)` |
| `:. l` | `= 80.058…` |
| `= 80.1\ 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 `(M)` is directly proportional to the cube of its length `(l)`.
An eel of this species has a length of 15 cm and a mass of 675 grams.
What is the expected length of a Giant moray eel with a mass of 3.125 kg? Give your answer to one decimal place. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`25\ text{cm}`
`M prop l^3`
`M = kl^3`
`text(Find)\ k:`
| `675` | `= k xx 15^3` |
| `k` | `= 675/15^3` |
| `= 0.2` |
`text(Find)\ \ l\ \ text(when)\ \ M = 3125:`
| `3125` | `= 0.2 xx l^3` |
| `l^3` | `= 3125/0.2` |
| `:. l` | `= root3(15\ 625)` |
| `= 25\ text{cm}` |
Jacques is a marine biologist and finds that the mass of a crab `(M)` is directly proportional to the cube of the diameter of its shell `(d)`.
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. (3 marks)
--- 6 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 height above the ground, in metres, of a person’s eyes varies directly with the square of the distance, in kilometres, that the person can see to the horizon.
A person whose eyes are 1.6 m above the ground can see 4.5 km out to sea.
How high above the ground, in metres, would a person’s eyes need to be to see an island that is 15 km out to sea? Give your answer correct to one decimal place. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`17.8\ text(m)\ \ text{(to 1 d.p.)}`
`h prop d^2`
`h=kd^2`
`text(When)\ h = 1.6,\ d = 4.5`
| `1.6` | `= k xx 4.5^2` |
| `:. k` | `= 1.6/4.5^2` |
| `= 0.07901` `…` |
`text(Find)\ h\ text(when)\ d = 15`
| `h` | `= 0.07901… xx 15^2` |
| `= 17.777…` | |
| `= 17.8\ text(m)\ \ \ text{(to 1 d.p.)}` |