Question

Only need help with part (e)!! thank you!!

Answer 1

shown in the picture

Explanation:

shown in the picture

Answer 2

a) 9000 L

`\textsf{b) (i)} \quad \textsf{Tap }P: \quad (9000)/(x)\:\sf minutes`

`\textsf{b) (ii)} \quad \textsf{Tap }Q: \quad (9000)/(x+5)\:\sf minutes`

c) Proof below.

`\textsf{d)} \quad x=(-5+5√(31) )/(2)=11.41941...`

e) 5 hours 23 minutes

Explanation:

Part (a)

The aquarium can be modeled as a rectangular prism.

Note: 1 m³ = 1000 L

`\begin{aligned}\textsf{Volume of rectangular prism} & = \sf length * width * height\\\\\implies \textsf{Volume of aquarium} & = \sf 3\:m * 2.5\:m * 1.2\:m\\& = \sf 9 \:\:m^3\\& = \sf 9000\:L\end{aligned}`

Part (b)

`\boxed{ \sf Time = (Volume)/(rate)}`

`\textsf{(i)} \quad \textsf{Tap }P: \quad (9000)/(x)\:\sf minutes`

`\textsf{(ii)} \quad \textsf{Tap }Q: \quad (9000)/(x+5)\:\sf minutes`

Part (c)

4 hours = 4 × 60 minutes = 240 minutes

If it takes 4 hours longer to fill the aquarium with water using tap P as compared to tap Q, then:

`\begin{aligned}\textsf{Tap }P - 240 & = \textsf{Tap }Q\\\\\implies (9000)/(x) -240 & = (9000)/(x+5)\\\\(9000 - 240x)/(x) & = (9000)/(x+5)\\\\(9000-240x)(x+5) & =9000x\\\\9000x + 45000 -240x^2-1200x & = 9000x\\\\-240x^2-1200x+45000 & =0\\\\-120(2x^2+10x-375) & =0\\\\2x^2+10x-375 & =0\end{aligned}`

Part (d)

Quadratic Formula

`x=(-b \pm √(b^2-4ac) )/(2a)\quad\textsf{when }\:ax^2+bx+c=0`

Therefore:

`a=2, \quad b=10, \quad c=-375`

Substitute these values into the formula and solve for x:

`\implies x=(-10 \pm √(10^2-4(2)(-375)) )/(2(2))`

`\implies x=(-10 \pm √(100+3000) )/(4)`

`\implies x=(-10 \pm √(3100) )/(4)`

`\implies x=(-10 \pm √(100 \cdot 31) )/(4)`

`\implies x=(-10 \pm √(100)√(31) )/(4)`

`\implies x=(-10 \pm 10√(31) )/(4)`

`\implies x=(-5\pm 5√(31) )/(2)`

As the rate is positive only,

`\implies x=(-5+5√(31) )/(2)=11.41941...\sf \:litres\:per\:minute`

Part (e)

If the rate that tap P fills the aquarium with water is x litres per min, and the rate the tap Q fills the aquarium with water is (x+5) litres per min, the rate that both taps fill the aquarium is:

`\begin{aligned}\implies \sf Rate & = x+(x+5)\\& = 2x + 5\\& = 2\left((-5+5√(31) )/(2)\right)+5\\& = -5+5√(31)+5\\& = 5√(31)\:\: \sf litres\:per\:min\end{aligned}`

Therefore, if both taps are turned on together, the time it takes to fill the 9000L aquarium is:

`\implies \sf Time & = (9000)/(5√(31) )\:\: \sf minutes \end{aligned}`

`\implies \sf Time & =323.2895436... \:\: \sf minutes \end{aligned}`

`\implies \sf Time & =5\:hours\:23.2895436... \:\: \sf minutes \end{aligned}`

`\implies \sf Time = 5\: h\: 23\: min`

Note we really should round up to the nearest minute, even though 23.289... rounded is 23, as if we round down, the aquarium will not quite be filled with water.