A cylindrical tank, with height of 15 m and diameter 4m, is being filled with gasoline at a rate of 0.5m^3 / min.
a) At what rate is the fluid level in the tank rising? Express your answer in terms of pi.
b) About how long will it take to fill the tank? Express your answer to the nearest minute.
If the tank is sitting on its end,
volume= h*PIr^2
dvolume/dt= dh/dt * PI r^2
solve for dh/dt, you know dV/dt
b. volume= ratevolume*time
PI r^2*15=.5 * t solve for time.
Your teacher is too easy.
could you expand and explain further please? i just want to check my answer
To answer these questions, we need to use the formula for the volume of a cylinder, which is V = πr^2h, where V is the volume, r is the radius, h is the height, and π is pi.
a) The rate at which the fluid level in the tank is rising is directly related to the rate of filling the tank. Since the fluid is being filled vertically, we need to find the rate at which the height is changing, which can be represented by dh/dt.
Given that the rate of filling the tank (dV/dt) is 0.5 m^3/min, we can use the formula for the volume to find dh/dt.
V = πr^2h
Differentiating both sides with respect to time (t), we get:
dV/dt = d(πr^2h)/dt
Since the radius (r) is half the diameter (d), we can substitute r = 2m into the equation, and differentiate it with respect to time (t):
dV/dt = d(π(2^2)h)/dt
0.5 = 4π(dh/dt)
dh/dt = 0.5 / (4π)
Therefore, the rate at which the fluid level in the tank is rising is 0.5 / (4π) m/min.
b) To find the time it takes to fill the tank, we need to first calculate the total volume of the tank.
V = πr^2h
Given the diameter (d) is 4 m, the radius (r) is half of that, so r = 2 m.
Also, the height (h) is 15 m.
Now, substitute these values into the equation to find the total volume:
V = π(2^2)(15)
V = π(4)(15)
V = 60π m^3
Next, divide the total volume of the tank by the rate at which it is being filled to find the time it takes to fill the tank:
Time = V / (dV/dt)
Time = (60π) / 0.5
Time = 120π minutes
Since the question asks for an approximate answer to the nearest minute, we get:
Time ≈ 377 minutes
Therefore, it will take approximately 377 minutes to fill the tank.