A state charges polluters an annual fee of $20 per ton for each ton of pollutant
emitted into the atmosphere, up to a maximum of 4,000 tons. No fees are charged
for emissions beyond the 4,000-ton limit. Write a piecewise de�nition of the fees F(x)
charged for the emission of x tons of pollutant in a year. What is the limit of F(x) as
x approaches 4,000 tons? As x approaches 8,000 tons?
F(x) is a continuous function governed by F(x)=20x for 0<x≤4000
=80000 for 4000<x
Note that
Lim x->4000- F(x)=Lim x->4000+ F(x) = 80000
See if you can find Lim x->8000 F(x).
Post your answer for confirmation if you wish.
80,000
80000
To write the piecewise definition of the fees F(x) charged for the emission of x tons of pollutant in a year, we need to consider the given conditions:
1. The fee is $20 per ton for each ton of pollutant emitted into the atmosphere.
2. The maximum limit for which fees are charged is 4,000 tons.
3. No fees are charged for emissions beyond the 4,000-ton limit.
Based on these conditions, we can write the piecewise definition of F(x) as follows:
F(x) =
- 20x, if 0 < x <= 4,000
- 0, if x > 4,000
Now let's analyze the limits as x approaches different values:
1. Limit of F(x) as x approaches 4,000 tons:
As x approaches 4,000 tons from the left side (0 < x < 4,000), the limit of F(x) is given by:
Lim (x->4,000-) F(x) = Lim (x->4,000-) (-20x) = -20 * 4,000 = -80,000
This means that as the emissions approach the 4,000-ton limit, the fees charged approach -$80,000.
As x approaches 4,000 tons from the right side (4,000 < x), F(x) becomes 0, since no fees are charged beyond the 4,000-ton limit.
2. Limit of F(x) as x approaches 8,000 tons:
As x approaches 8,000 tons from both the left side (0 < x < 8,000) and the right side (x > 8,000), F(x) becomes 0 since no fees are charged beyond the 4,000-ton limit.
Lim (x->8,000-) F(x) = Lim (x->8,000) F(x) = 0
Therefore, the limit of F(x) as x approaches 8,000 tons is 0, indicating that no fees are charged beyond the 4,000-ton limit.