Question

Ask Your Teacher A trough is 16 ft long and its ends have the shape of isosceles triangles that are 2 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 9 ft3/min, how fast is the water level rising when the water is 8 inches deep?

Answer 1

Answer:0.210 ft/min

Step-by-step explanation:

Given

Length of trough
`L=16 ft`

width of base
`b=2 ft`

height of triangle
`h=1 ft`

From Similar triangles property

`(4)/(2x)=(1)/(y)`

`2y=x`

volume of water in time t

`V=(1)/(2)* (2x\cdot y)\cdot`

`V=16xy`

`V=32y^2`

differentiating

`\frac{\mathrm{d} V}{\mathrm{d} t}=32* 2* y* \frac{\mathrm{d} y}{\mathrm{d} t}`

at
`y=8 in.\approx 0.667 ft`

`9=64* 0.667* \frac{\mathrm{d} y}{\mathrm{d} t}`

`\frac{\mathrm{d} y}{\mathrm{d} t}=(9)/(64* 0.667)`

`\frac{\mathrm{d} y}{\mathrm{d} t}=0.210 ft/min`