smc-1099-40-Sketch equations

Algebra, STD1 A3 2021 HSC 29

In a park the only animals are goannas and emus. Let `x` be the number of goannas and let `y` be the number of emus.

The number of goannas plus the number of emus in the park is 31. Hence `x + y = 31`.

Each goanna has four legs and each emu has two legs. In total the emus and goannas have 76 legs.

By writing another relevant equation and graphing both equations on the grid on the following page, find the number of goannas and the number of emus in the park. (4 marks)

--- 5 WORK AREA LINES (style=lined) ---

Number of goannas = _______________

Number of emus = _________________

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`text{7 goannas, 24 emus}`

Show Worked Solution

`text{Total goanna legs} = 4x`

♦♦♦ Mean mark 8%.

`text{Total emu legs} = 2y`

`4x + 2y`	`= 76`
`2x + y`	`= 38 \ …\ (1)`
`x + y`	`= 31 \ …\ (2)`

`text{Number of goannas} = 7`

`text{Number of emus} = 24`

Algebra, STD1 A3 2020 HSC 29

There are two tanks on a property, Tank A and Tank B. Initially, Tank A holds 1000 litres of water and Tank B is empty.

Tank A begins to lose water at a constant rate of 20 litres per minute.

The volume of water in Tank A is modelled by `V = 1000 - 20t` where `V` is the volume in litres and `t` is the time in minutes from when the tank begins to lose water.

On the grid below, draw the graph of this model and label it as Tank A. (1 mark)
Tank B remains empty until `t=15` when water is added to it at a constant rate of 30 litres per minute.
By drawing a line on the grid (above), or otherwise, find the value of `t` when the two tanks contain the same volume of water. (2 marks)
Using the graphs drawn, or otherwise, find the value of `t` (where `t > 0`) when the total volume of water in the two tanks is 1000 litres. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

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`text{T} text{ank} \ A \ text{will pass trough (0, 1000) and (50, 0)}`
`29 \ text{minutes}`
`45 \ text{minutes}`

Show Worked Solution

a. `text{T} text{ank} \ A \ text{will pass trough (0, 1000) and (50, 0)}`

♦ Mean mark part (a) 45%.

b. `text{T} text{ank} \ B \ text{will pass through (15, 0) and (45, 900)}`

♦♦ Mean mark part (b) 24%.

`text{By inspection, the two graphs intersect at} \ \ t = 29 \ text{minutes}`

c. `text{Strategy 1}`

♦♦♦ Mean mark part (c) 5%.

`text{By inspection of the graph, consider} \ \ t = 45`

`text{T} text{ank A} = 100 \ text{L} , \ text{T} text{ank B} =900 \ text{L} `

`:.\ text(Total volume = 1000 L when t = 45)`

`text{Strategy 2}`

`text{Total Volume}`	`=text{T} text{ank A} + text{T} text{ank B}`
`1000`	`= 1000 – 20t + (t – 15) xx 30`
`1000`	`= 1000 – 20t + 30t – 450 `
`10t`	`= 450`
`t`	`= 45 \ text{minutes}`