Respuesta :
Answer:
When the water level is 3 cm, the rate at which the water level is rising is 30.86 cm/s.
Step-by-step explanation:
We first have to relate the height H of the pyramid with the volume V.
The volume of the pyramid is V=Sh/3, being S: area of the base square, and h: height of the pyramid.
The base is changing as the pyramid is filled with water.
Then, the volume for every height is:
[tex]V(t)=[L(t)]^2h(t)/3[/tex]
The side of the square will grow linearly with the height. They start at H=0 and L=0 at the bottom of the pyramid, and they end at H=10 and L=3.
So we have:
[tex]L=(3/10)h=0.3h[/tex]
If we replace in the volume equation we have:
[tex]V=L^2h/3=(0.3h)^2h/3=0.09h^3/3=0.03h^3[/tex]
We know that the rate of variation of the volume in time is contant and equal to 25 cm3:
[tex]dV/dt=25[/tex]
We can derive the volume equation to calculate the variation of h in time:
[tex]dV/dt=3(0.03)h^2*(dh/dt)=0.09h^2*(dh/dt)=25\\\\dh/dt=(25/0.09)h^{-2}=278h^{-2}[/tex]
The rate of variation of the height in time is dh/dt=278h^(-2). The units are cm/s.
When the water level is 3 cm, the rate at which the water level is rising is:
[tex]h=3\\\\dh/dt=278*3^{-2}=278*(1/9)=30.86[/tex]
