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

Explanation:

Given

Length of trough $L=16 ft$

width of base $b=2 ft$

height of triangle$h=1 ft$

From Similar triangles property

$\frac{4}{2x}=\frac{1}{y}$

$2y=x$

volume of water in time t

$V=\frac{1}{2}\times (2x\cdot y)\cdot$

$V=16xy$

$V=32y^2$

differentiating

$\frac{\mathrm{d} V}{\mathrm{d} t}=32\times 2\times y\times \frac{\mathrm{d} y}{\mathrm{d} t}$

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

$9=64\times 0.667\times \frac{\mathrm{d} y}{\mathrm{d} t}$

$\frac{\mathrm{d} y}{\mathrm{d} t}=\frac{9}{64\times 0.667}$

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