Question

the increase in a persons body temperature ( t(t) ), above ( 98.6^circ\text{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 = 0 ) means that the persons temperature will approach ( 98.6^circ\text{f} ) as time elapses.
• the horizontal asymptote of ( y = 0 ) means that the persons temperature will approach ( 0^circ\text{f} ) as time elapses.
• the horizontal asymptote of ( y = 4 ) means that the persons temperature will approach ( 102.6^circ\text{f} ) as time elapses.
• the horizontal asymptote of ( y = 4 ) means that the persons temperature will approach ( 4^circ\text{f} ) as time elapses

Explanation:

Step1: Find Horizontal Asymptote

For a rational function \( T(t)=\frac{4t}{t^{2}+1} \), when the degree of the numerator (\(1\)) is less than the degree of the denominator (\(2\)), the horizontal asymptote is \( y = 0 \).

Step2: Interpret the Asymptote

\( T(t) \) represents the increase in body temperature above \( 98.6^\circ\text{F} \). As \( t\to\infty \) (time elapses), \( T(t)\to0 \), meaning the increase approaches \( 0 \), so the person's temperature approaches \( 98.6^\circ\text{F} \) (since initial normal is \( 98.6^\circ\text{F} \) and increase approaches \( 0 \)).

Answer:

The horizontal asymptote of \( y = 0 \) means that the person’s temperature will approach \( 98.6^\circ\text{F} \) as time elapses.