The percentage of deaths caused by smoking related heart disease is represented by the function
where is the total number of heart disease related deaths at a given time.
a) Find the equation of the horizontal asymptote for this function.
b) As the number of heart disease related deaths increases, the percent of smoking related heart disease
deaths approaches

Ask by Barnett Boyd. in the United States

Nov 22,2024

Question

The percentage of deaths caused by smoking related heart disease is represented by the function
where is the total number of heart disease related deaths at a given time.
a) Find the equation of the horizontal asymptote for this function.
b) As the number of heart disease related deaths increases, the percent of smoking related heart disease
deaths approaches

Ask by Barnett Boyd. in the United States

Nov 22,2024

UpStudy AI · Accepted Answer

To analyze the function $$ P(x) = \frac{65(x-1)}{x} $$, we can find the horizontal asymptote and determine the behavior of the function as $$ x $$ approaches infinity.

### a) Finding the horizontal asymptote

To find the horizontal asymptote, we need to evaluate the limit of $$ P(x) $$ as $$ x $$ approaches infinity:

$$
\lim_{x \to \infty} P(x) = \lim_{x \to \infty} \frac{65(x-1)}{x}
$$

We can simplify the expression:

$$
P(x) = \frac{65(x-1)}{x} = \frac{65x - 65}{x} = 65 - \frac{65}{x}
$$

Now, taking the limit as $$ x $$ approaches infinity:

$$
\lim_{x \to \infty} P(x) = 65 - \lim_{x \to \infty} \frac{65}{x} = 65 - 0 = 65
$$

Thus, the horizontal asymptote is:

$$
y = 65
$$

### b) Behavior as the number of heart disease related deaths increases

As the number of heart disease related deaths $$ x $$ increases, the percentage of deaths caused by smoking-related heart disease approaches the horizontal asymptote we found:

$$
\text{As } x \to \infty, P(x) \to 65
$$

Therefore, the percent of smoking-related heart disease deaths approaches **65%**.
a) The horizontal asymptote is $$ y = 65 $$. b) The percent of smoking-related heart disease deaths approaches 65%.

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Solución

a) Finding the horizontal asymptote

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