A 12,000 liter tank filled to capacity at t=0 drains at rate r(t), measured in liters per hour, where r is defined as piece-wise function:
r(t)={ (600t)/(t+3), [0,5]
{1000e^(-0.2t), (5,infinity)
Is r continuous at t=5? Show work that lead to you answer.
at t=5,
600t/(t+3) = 3000/8 = 375
1000e^(-.2t) = 1000/e = 367.8
Actually, the computation isn't really necessary, since r(t) is rational at t=5, and e^x is irrational for any rational x.
what does your answer mean exactly?
it means that the two piecewise functions don't have the same value at t=5, so r(t) cannot be continuous.
To determine whether the function r(t) is continuous at t = 5, we need to check if the two pieces of the function match up at that point. In other words, we need to compare the values of r(t) as t approaches 5 from both the left and the right side.
First, let's calculate the value of r(t) as t approaches 5 from the left side (t < 5). For this case, we'll use the first piece of the function:
r(t) = (600t)/(t+3)
As t approaches 5 from the left side, we substitute t = 5 into the function:
r(5-) = (600 * 5) / (5 + 3)
= 3000 / 8
= 375 liters per hour
Next, let's calculate the value of r(t) as t approaches 5 from the right side (t > 5). For this case, we'll use the second piece of the function:
r(t) = 1000e^(-0.2t)
As t approaches 5 from the right side, we substitute t = 5 into the function:
r(5+) = 1000e^(-0.2 * 5)
= 1000e^(-1)
≈ 367.88 liters per hour
Since the value of r(5-) is not the same as r(5+), we can conclude that the function r(t) is not continuous at t = 5.
To further visualize this, we can plot the graph of r(t) and observe a jump/discontinuity at t = 5, where the two pieces of the function join.