Respuesta :
The rate of water level rising when the water is 30 cm deep will be 1/30 m/min.
What is volume?
The term “volume” refers to the amount of three-dimensional space taken up by an item or a closed surface. It is denoted by V and its SI unit is in cubic cm.
The complete question is;
"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 a height of 50cm. If the trough is being filled with water at the rate of 0.2 m3/min, how fast is the water level rising when the water is 30cm deep?"
b1 is the width of the water at a height at the bottom
b2 is the width of the water at the height at the top
The length of the trapezoid is L
The volume of the trapezoid is found as;
[tex]\rm V = 0.5(b_1 + b_2)hL[/tex]
The breadth rises by 1 as the height does, therefore which implies
[tex]\rm \frac{dh}{dt} =\frac{dw}{dt}[/tex]
The water's breadth at the combined height is [0.3 + h]
[tex]\rm V = 5h(0.3 +(0.3 + h))\\\\ V = 3h + 5h^2[/tex]
After differentiation we get;
[tex]\rm \frac{dv }{dt} =3 \frac{dh}{dt} +10 h \frac{dh}{dt} \\\\\ \frac{ \frac{dv }{dt}}{3+10h}=\frac{dh}{dt} \\\\\ \frac{dv }{dt}= 0.2 \\\\\ h = 0.3 \\\\ \frac{dh}{dt} = \frac{1}{30} m/min[/tex]
Hence the rate of water level rising when the water is 30 cm deep will be 1/30 m/min.
To learn more about the volume, refer to https://brainly.com/question/1578538.
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