SOLUTION: 𝐏𝐫𝐨𝐛𝐥𝐞𝐦 𝟒

Question 1170427: 𝐏𝐫𝐨𝐛𝐥𝐞𝐦 𝟒

(b) During a nationwide program to immunize the population against a new strain of the flu, public health officials determined that the cost of inoculating x% of the susceptible population
1.85x
would be approximately C(x) = million dollars. x
(i) What would it cost to providing immunization to the first 20% of the susceptible population?
(ii) What would it cost to providing immunization to the next 30% of the susceptible population?
(iii) Suppose 17 million dollars are available for providing immunization. What percentage of the susceptible population will not receive immunization?
(iv) If money was not a problem will they be able to providing immunization to the entire susceptible population?

(c) Determine the values of x for which the function f

is discontinuous.

Answer by CPhill(2138) (Show Source): You can put this solution on YOUR website!
Let's break down each part of the problem.
**(b) Immunization Costs**
We are given the cost function:
$$C(x) = \frac{1.85x}{100-x}$$
where $x$ is the percentage of the susceptible population immunized, and $C(x)$ is in million dollars.
**(i) Cost for the first 20%:**
We need to find $C(20)$.
$$C(20) = \frac{1.85(20)}{100-20} = \frac{37}{80} = 0.4625$$
So, it would cost 0.4625 million dollars, or $462,500.
**(ii) Cost for the next 30%:**
If we immunize the first 20%, then we need to immunize from 20% to 50%.
First, find the cost of immunizing 50%:
$$C(50) = \frac{1.85(50)}{100-50} = \frac{92.5}{50} = 1.85$$
The cost for the next 30% is $C(50) - C(20)$:
$$1.85 - 0.4625 = 1.3875$$
So, it would cost 1.3875 million dollars, or $1,387,500.
**(iii) Percentage not immunized with $17 million:**
We need to find $x$ such that $C(x) = 17$:
$$17 = \frac{1.85x}{100-x}$$
$$17(100-x) = 1.85x$$
$$1700 - 17x = 1.85x$$
$$1700 = 18.85x$$
$$x = \frac{1700}{18.85} \approx 90.1856764$$
So, approximately 90.19% of the population can be immunized.
The percentage not immunized is $100 - x$:
$$100 - 90.19 \approx 9.81$$
Approximately 9.81% of the population will not receive immunization.
**(iv) Immunization for the entire population:**
If we want to immunize 100% of the population, we need to find $C(100)$:
$$C(100) = \frac{1.85(100)}{100-100} = \frac{185}{0}$$
This is undefined, as we cannot divide by zero.
Therefore, it is impossible to immunize the entire susceptible population.
**(c) Discontinuity of f(x)**
You didn't provide the function f(x). However, I can explain how to find the discontinuities of a function.
**Finding Discontinuities:**
* **Rational Functions:** If $f(x)$ is a rational function (a fraction where the numerator and denominator are polynomials), then the function is discontinuous where the denominator is zero.
* **Piecewise Functions:** If $f(x)$ is a piecewise function, then the function is discontinuous where the pieces do not meet.
* **Other Functions:** If $f(x)$ involves other functions like logarithms or square roots, then the function is discontinuous where those functions are undefined.
**Example:**
Let's say $f(x) = \frac{x+1}{x-2}$.
To find the discontinuities, we set the denominator to zero:
$x - 2 = 0$
$x = 2$
Therefore, $f(x)$ is discontinuous at $x = 2$.
**Please provide the function f(x) so I can determine its discontinuities.**

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