Question

the increase in a persons body temperature t(t), above 98.6°f, can be modeled by the function t(t) = \frac{4t}{t^2 + 1}, where t represents time elapsed. what is the meaning of the horizontal asymptote for this function?

the horizontal asymptote of y = 4 means that the persons temperature will approach 102.6°f as time elapses.

the horizontal asymptote of y = 0 means that the persons temperature will approach 98.6°f as time elapses.

the horizontal asymptote of y = 4 means that the persons temperature will approach 4°f as time elapses.

the horizontal asymptote of y = 0 means that the persons temperature will approach 0°f as time elapses.

Explanation:

Step1: Find Horizontal Asymptote

For a rational function \( T(t)=\frac{4t}{t^{2}+1} \), the degree of the numerator (\( n = 1 \)) and the degree of the denominator (\( m = 2 \)). When \( n

Step2: Interpret the Asymptote

\( T(t) \) represents the increase in body temperature above \( 98.6^\circ\text{F} \). If the horizontal asymptote is \( y = 0 \), it means the increase in temperature approaches \( 0 \), so the person’s temperature approaches \( 98.6^\circ\text{F}+0 = 98.6^\circ\text{F} \) as time \( t \) elapses.

Answer:

The horizontal asymptote of \( y = 0 \) means that the person's temperature will approach \( 98.6^\circ \text{F} \) as time elapses. (The option with this description, e.g., the second option in the given choices)