A water tank has a square base of area 5 square meters. Initially the tank contains 70 cubic meters. Water leaves the tank, star
ting at t=0, at the rate of 2 + 4 t cubic meters per hour. Here t is the time in hours. What is the depth of water remaining in the tank after 3 hours
1 answer:
Answer:
![9.2\ \text{m}](https://tex.z-dn.net/?f=9.2%5C%20%5Ctext%7Bm%7D)
Step-by-step explanation:
= Surface area of base = 5 square meters
Volume of water in tank = 70 cubic meters
The rate at which the volume is reducing is
![\dfrac{dV}{dt}=2+4t\\\Rightarrow dV=2+4tdt](https://tex.z-dn.net/?f=%5Cdfrac%7BdV%7D%7Bdt%7D%3D2%2B4t%5C%5C%5CRightarrow%20dV%3D2%2B4tdt)
Integrating from
to ![t=3](https://tex.z-dn.net/?f=t%3D3)
![V=\int^3_0(2+4t)dt\\\Rightarrow V=2t+2t^2|_0^3\\\Rightarrow V=2\times 3+2\times 3^2-0\\\Rightarrow V=24](https://tex.z-dn.net/?f=V%3D%5Cint%5E3_0%282%2B4t%29dt%5C%5C%5CRightarrow%20V%3D2t%2B2t%5E2%7C_0%5E3%5C%5C%5CRightarrow%20V%3D2%5Ctimes%203%2B2%5Ctimes%203%5E2-0%5C%5C%5CRightarrow%20V%3D24)
Volume of water remaining in the tank is ![70-24=46\ \text{m}^3](https://tex.z-dn.net/?f=70-24%3D46%5C%20%5Ctext%7Bm%7D%5E3)
Suface area of base
depth = Volume
![5\times d=46\\\Rightarrow d=\dfrac{46}{5}\\\Rightarrow d=9.2\ \text{m}](https://tex.z-dn.net/?f=5%5Ctimes%20d%3D46%5C%5C%5CRightarrow%20d%3D%5Cdfrac%7B46%7D%7B5%7D%5C%5C%5CRightarrow%20d%3D9.2%5C%20%5Ctext%7Bm%7D)
The depth of the water remaining in the tank is
.
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