Posted by asdf on Saturday, December 1, 2012 at 4:19pm.
A water-trough is 10m long and has a cross-section which is the shape of an isosceles trapezoid
that is 30cm wide at the bottom, 80cm wide at the top, and has height 50cm. If the trough is being fi lled with water at the rate of 0.2 m3/min, how fast is the water level rising when the water is 30cm deep?
The steps I took were as follows
b1 is the width of the water at height 0
b2 is the width of the water at height h
knowing the volume of a trapezoid is
V = 0.5(b1 + b2)hL
where L is the length of the trapezoid.
As the height increases by 1 so does the width so that means dh/dt = dw/dt
Now I tried solving for h in meters.
V = 5h(0.3 +(0.3 + h))
[0.3 + h is the width of the water at the combined height]
V = 3h + 5h^2
differentiate by d/dt
dV/dt = 3dh/dt + 10h dh/dt
(dV/dt)/(3 + 10h) = dh/dt
solve for dV/dt = 0.2 and h = 0.3
1/30 m/min = dh/dt
The answer is 1/3 cm/min but when I convert this to cm I get 10/3 cm/min.
What exactly am I doing wrong here?
- calculus - Damon, Saturday, December 1, 2012 at 6:56pm
well first of all think easy way
for small changes (as dh ---> 0)
change in volume = surface area * change in height
dV = A dh
Now do the problem (the surface is a rectangle)
we know the length, 10 meters
what is the width when h = 30 ???
linear in depth (straight sides)
w = .30 + [(.80-.30) /.50] h
w = .3 + h
so for the area of the surface
A = 10(.3+h)
when h = .3, A = 10(.6) = 6 m^2
dV = A dh
dV/dt = A dh/dt
.2 m^3/min = 6 m^2 dh/dt
dh/dt = (.2/6) meters/min
= .03333 meters/min
= 3 1/3 cm /min
Answer This Question
More Related Questions
- Calculus - A water trough is 9 m long and has a cross-section in the shape of an...
- Differential Calculus - A water trough is 10 m long and a cross section has the ...
- Calculus Help Please!!! - A water trough is 5 m long and has a cross-section in ...
- Calculus - A water trough is 5 m long and has a cross-section in the shape of an...
- calc - A water trough is 7 m long and has a cross-section in the shape of an ...
- Calculus - A water trough is 6 m long, and its cross-section has the shape of an...
- math - A water trough is 4 m long and its cross-section is an isosceles ...
- maths - A large canal will have a cross-section in the shape of an isosceles ...
- maths - the cross section of a canal is a trapezium in shape .if the canal is ...
- Calculus - A trough is 5 meters long, 1 meters wide, and 4 meters deep. The ...