Question 1175631: A consultant has issued an environmental report on the cost of cleaning up a property that was previously the site of a chemical factory. Costs can increase dramatically depending on the percent of pollutants that needs to be removed. Her report gives the cost, C, in dollars, of removing p% of the pollutants from the site as C(p)=50000/100-p
a) Determine the average rate of change in the cost of cleaning up between 15% and 20% of the pollutants on the property.
b)Estimate the rate of change in the cost of cleaning up 95% of the pollutants on the property.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step.
**a) Average Rate of Change Between 15% and 20%**
* **Formula:** The average rate of change of a function C(p) between p1 and p2 is given by:
* (C(p2) - C(p1)) / (p2 - p1)
* **Calculations:**
* C(p) = 50000 / (100 - p)
* p1 = 15, p2 = 20
* C(15) = 50000 / (100 - 15) = 50000 / 85 ≈ 588.24
* C(20) = 50000 / (100 - 20) = 50000 / 80 = 625
* Average rate of change = (625 - 588.24) / (20 - 15)
* Average rate of change = 36.76 / 5 ≈ 7.35
* **Answer:** The average rate of change in the cost of cleaning up between 15% and 20% of the pollutants is approximately $7.35 per percentage point.
**b) Estimate the Rate of Change at 95%**
* **Concept:** To estimate the instantaneous rate of change (or the derivative) at a point, we can use a very small change in the input value.
* **Calculations:**
* C(p) = 50000 / (100 - p)
* p = 95
* Let's use a small change in p, let's say 0.001, so delta_p = 0.001
* C(95) = 50000/(100-95) = 50000/5 = 10000
* C(95.001) = 50000/(100-95.001) = 50000/4.999 = 10002.0004
* Rate of change = (C(95.001)-C(95))/0.001
* Rate of change = (10002.0004-10000)/0.001 = 2.0004/0.001 = 2000.4
* **Answer:** The estimated rate of change in the cost of cleaning up 95% of the pollutants is approximately $2000.40 per percentage point.
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