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. (a) 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 $V = 1000 - 20i$, where $V$ is the volume in litres and $i$ 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. (b) Tank B remains empty until $i = 15$, when water is added to it at a constant rate of 30 litres per minute. By drawing a line on the grid on the previous page, or otherwise, find the value of $i$ when the two tanks contain the same volume of water. (c) Using the graphs drawn, or otherwise, find the value of $i$ (where $i > 0$) when the total volume of water in the two tanks is 1000 litres.

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There are two tanks on a property, Tank A and Tank B - HSC - SSCE Mathematics Advanced - Question 11 - 2020 - Paper 1

Question 11

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. (a) Tank A begins to lose water at a constan... show full transcript

Worked Solution & Example Answer:There are two tanks on a property, Tank A and Tank B - HSC - SSCE Mathematics Advanced - Question 11 - 2020 - Paper 1

Step 1

Part (a): Graph the Volume of Tank A

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Answer

To graph the volume of Tank A, we start with the equation $V = 1000 - 20i$ . This equation shows that initially (when $i=0$ ), the volume is 1000 litres and decreases at a rate of 20 litres per minute.

We can calculate a few points:

At $i = 0$ : $V = 1000 - 20(0) = 1000$ litres
At $i = 5$ : $V = 1000 - 20(5) = 900$ litres
At $i = 10$ : $V = 1000 - 20(10) = 800$ litres
At $i = 15$ : $V = 1000 - 20(15) = 700$ litres
At $i = 50$ : $V = 1000 - 20(50) = 0$ litres

Plot these points on the grid and connect them with a straight line, labeling it as "Tank A".

Step 2

Part (b): Find the Value of $i$ When Tanks Contain Equal Volume

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Answer

For Tank B, we know it starts filling up at $i = 15$ minutes at a rate of 30 litres per minute. Thus, the volume of Tank B can be represented as:

$V_B = 30(i - 15)$ for $i ext{ } ext{ } (i ext{ } ext{ } 15)$$

To find when both tanks have the same volume, we set the two equations equal to each other:

$1000 - 20i = 30(i - 15)$

Solving this equation gives:

Expand and simplify: $1000 - 20i = 30i - 450$
Combine like terms: $1000 + 450 = 30i + 20i$ $1450 = 50i$
Divide by 50: $i = 29$

Thus, $i = 29$ minutes.

Step 3

Part (c): Value of $i$ When Total Volume is 1000 Litres

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Answer

To find when the total volume of both tanks is 1000 litres, we need to consider when:

$V_A + V_B = 1000$

From part (a), we know: $V_A = 1000 - 20i$

From part (b), we have: $V_B = 30(i - 15)$ (for $i ext{ } ext{ } 15$ )

Setting them equal: $1000 - 20i + 30(i - 15) = 1000$

Expanding this gives: $1000 - 20i + 30i - 450 = 1000$ $10i - 450 = 0$ $10i = 450$ $i = 45$

Thus, the value of $i$ when the total volume is 1000 litres is $i = 45$ minutes.

There are two tanks on a property, Tank A and Tank B - HSC - SSCE Mathematics Advanced - Question 11 - 2020 - Paper 1

Worked Solution & Example Answer:There are two tanks on a property, Tank A and Tank B - HSC - SSCE Mathematics Advanced - Question 11 - 2020 - Paper 1

Part (a): Graph the Volume of Tank A

Part (b): Find the Value of $i$ When Tanks Contain Equal Volume

Part (c): Value of $i$ When Total Volume is 1000 Litres

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