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)
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`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.)}` |
Taylor discovers that for a Spotted stingray, its mass is directly proportional to the square of its wingspan.
One Spotted stingray has a wingspan of 60 cm and a mass of 5400 grams.
What is the expected wingspan of a Spotted stingray with a mass of 9.6 kg? (3 marks)
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`80.0\ text{cm}`
`text(Mass) prop text(wingspan)^2\ \ =>\ \ m = kw^2`
`text(Find)\ k:`
| `5400` | `= k xx 60^2` |
| `k` | `= 5400/60^2= 1.5` |
`text(Find)\ w\ text(when)\ \ m = 9600:`
| `9600` | `= 1.5 xx w^2` |
| `w^2` | `= 9600/1.5=6400` |
| `:. w` | `= 80\ text{cm}` |
The stopping distance of a motor bike, in metres, is directly proportional to the square of its speed in km/h, and can be represented by the equation
`text{stopping distance}\ = k xx text{(speed)}^2`
where `k` is the constant of variation.
The stopping distance for a motor bike travelling at 40 km/h is 16 m.
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a. `k=0.01`
b. `64.0\ text{m}`
a. `text{stopping distance}\ = k xx text{(speed)}^2`
| `16` | `=k xx 40^2` | |
| `k` | `=16/40^2=0.001` |
b. `text{Find stopping distance}\ (d)\ text{when speed = 80 km/h:}`
`d=0.01 xx 80^2=64.0\ text{m}`
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.
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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 braking distance}\ (d)\ text{when speed = 90 km/h:}`
| `d` | `=0.008 xx 90^2` | |
| `=64.8\ text{m}` |
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)
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`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` | `= 80.069…` |
| `= 80.1\ 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)
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`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.)}` |