If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume of water remaining in the tank after t minutes as
V(t)=100,000(1-(t/50))^2, 0<=t<=60. Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t.
a)y=(-100000/50)+(100000t/1)
b) y=(-4000/1)+(160t/1)
c)y=(-100000/50)+(80t/1)
d)y= (200000/50)-(200000t/2500)
e)(-4000/1)+(80t/1)
y = dV/dt
= 100,000 * 2(1 - t/50)(-1/50)
= -4000 (1 - t/50)
= -4000 + 80t
This is (e) with all the noise removed
To find the rate at which the water is flowing out of the tank, we need to differentiate the volume function V(t) with respect to time t.
Using the power rule of differentiation, the derivative of V(t) is:
V'(t) = 2 * 100,000 * (1 - (t/50)) * (-1/50)
Simplifying the expression gives:
V'(t) = -2,000,000 * (1 - (t/50)) * (1/50)
Now, let's simplify it further:
V'(t) = -2,000,000/50 * (1 - (t/50))
V'(t) = -40,000 * (1 - (t/50))
Therefore, the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t is approximately represented as:
y = (-40,000 * (1 - (t/50))) or simplified as y = (-40,000 + 800t)
To find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t), we need to find the derivative of the volume function V(t) with respect to t.
Given that V(t) = 100,000(1 - t/50)^2, we can use the chain rule to differentiate the function.
Step 1: Apply the power rule for differentiation.
dV(t)/dt = 100,000 * 2 * (1 - t/50) * (-1/50)
Step 2: Simplify the expression.
dV(t)/dt = -2,000,000 * (1 - t/50) / 50
Step 3: Further simplify the expression.
dV(t)/dt = -40,000 + 40,000t/50
Step 4: Divide by 1 to get the final form.
dV(t)/dt = -40,000 + 800t
Therefore, the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t is given by:
y = -40,000 + 800t
The correct option is e) (-4000/1) + (80t/1).