Hi all,
There is no indication of base so am wondering how to solve this sum. Thank you for taking the time.
Hello, fleurdel!
18. There are two tanks filled with water.
They are being emptied by two different taps, each with a constant flow of water.
Tank A is 5 cm taller than Tank B.
Code:* - - - * |: : : :| | : : : | * - - - - * |: : : :| |: : : : :| x | : : : | | : : : : | |: : : :| |: : : : :| x-5 | : : : | | : : : : | |: : : :| |: : : : :| *-------* *---------* A B
The tap of Tank A was opened at 7:00 a.m.
Tank A was empty at 1:00 p.m.
The tap of Tank B was opened at 8:30 a.m.
Tank B was empty at 12:30 p.m.
The water level of both tanks were equal at 11 a.m.
What is the height of tank A?
$\displaystyle \text{Let }a\text{ = rate of flow of the tap on Tank A, relative to the height of the water,}$
. . $\displaystyle \text{measured in cm/hour.}$
$\displaystyle \text{Let }b\text{ = rate of flow of the tap on Tank B, relative to the height of the watr,}$
. . $\displaystyle \text{measured in cm/hour.}$
$\displaystyle \text{In 6 hours, the height of Tank A went from }x\text{ cm to 0 cm.}$
. . $\displaystyle \text{We have: }\:6a \,=\,x$ .[1]
$\displaystyle \text{In 4 hours, the height of Tank B went from }x-5\text{ cm to 0 cm.}$
. . $\displaystyle \text{We have: }\:4b \,=\,x-5$ .[2]
Substitute [1] into [2]: .$\displaystyle 4b \:=\:6a - 5 \quad\Rightarrow\quad 6a - 4b \:=\:5$ .[3]
At ll:00 a.m. the two tanks had the same height.
Tank A: in 4 hours, the level dropped $\displaystyle 4a\text{ cm.}$
. . The height was: .$\displaystyle x - 4a\text{ cm.}$
Tank B: in 2.5 hours, the level dropped $\displaystyle \tfrac{5}{2}b\text{ cm.}$
. . The height was: .$\displaystyle (x-5)-\tfrac{5}{2}b\text{ cm.}$
We have: .$\displaystyle x - 4a \:=\:(x-5) - \tfrac{5}{2}b \quad\Rightarrow\quad 8a - 5b \:=\:10$ .[4]
Solve [3] and [4]: .$\displaystyle \begin{Bmatrix}6a - 4b &=& 5 \\ 8a - 5b &=& 10 \end{Bmatrix} \quad\Rightarrow\quad \begin{Bmatrix}a &=& \frac{15}{2} \\ b &=& 10 \end{Bmatrix}$
Substitute into [1]: .$\displaystyle x \:=\:6a \:=\:6\left(\tfrac{15}{2}\right) \:=\:45$
Therefore, Tank A is 45 cm high.