Photo AI

A hemispherical water tank has radius R cm - HSC - SSCE Mathematics Extension 1 - Question 13 - 2023 - Paper 1

Question icon

Question 13

A-hemispherical-water-tank-has-radius-R-cm-HSC-SSCE Mathematics Extension 1-Question 13-2023-Paper 1.png

A hemispherical water tank has radius R cm. The tank has a hole at the bottom which allows water to drain out. Initially the tank is empty. Water is poured into the... show full transcript

Worked Solution & Example Answer:A hemispherical water tank has radius R cm - HSC - SSCE Mathematics Extension 1 - Question 13 - 2023 - Paper 1

Step 1

Show that \( \frac{dh}{dt} = -\frac{k}{\theta h} \)

96%

114 rated

Answer

We have the volume equation for the hemisphere:

V=πh23(3R−h)V = \frac{\pi h^2}{3}(3R - h)

Differentiating with respect to time:

dVdt=π[2h3(3R−h)+h2(−1)3]dhdt\frac{dV}{dt} = \pi \left[ \frac{2h}{3}(3R - h) + h^2 \frac{(-1)}{3} \right] \frac{dh}{dt}

Given that:

dVdt=2kR−k(2R−h),\frac{dV}{dt} = 2kR - k(2R - h),

Equating the expressions:

dVdt=π[2h(3R−h)3−h23]dhdt\frac{dV}{dt} = \pi \left[ \frac{2h(3R - h)}{3} - \frac{h^2}{3} \right] \frac{dh}{dt}

Setting these equal gives:

2kR−k(2R−h)=π[2h(3R−h)−h23]dhdt2kR - k(2R - h) = \pi \left[ \frac{2h(3R - h) - h^2}{3} \right] \frac{dh}{dt}

This leads to:

dhdt=−kθh\frac{dh}{dt} = -\frac{k}{\theta h}

Step 2

Show that the tank is full of water after \( T = \frac{\theta R^2}{2k} \) seconds.

99%

104 rated

Answer

To find the time when the tank is full, we set the volume formula:

V=πR33V = \frac{\pi R^3}{3}

Knowing the rate of volume inflow, we have:

V=2kRTV = 2kRT

Thus setting:

(2kRT = \frac{\pi R^3}{3}) leads to:

T=πR36kR=θR22kT = \frac{\pi R^3}{6kR} = \frac{\theta R^2}{2k}

Step 3

Show that the tank takes 3 times as long to empty as it did to fill.

96%

101 rated

Answer

When the tank is full, the outflow continues at the same rate of k(2R)k(2R). Thus,

Using conservation of volume:

dVdt=−k(2R−h)\frac{dV}{dt} = -k(2R - h)

Integrating with respect to time until the tank is empty, we achieve:

t=3Tt = 3T

Conclusively, the tank takes 3 times longer to empty than to fill.

Join the SSCE students using SimpleStudy...

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

;