Answer :
We are given the temperature function
[tex]$$
T(t) = -0.01t^2 + 0.218t + 97.6,
$$[/tex]
where [tex]\( t \)[/tex] is in hours. Since the coefficient [tex]\( a = -0.01 \)[/tex] is negative, the quadratic function opens downward and therefore reaches a maximum at its vertex.
Step 1. Find the time at which the maximum occurs.
The time corresponding to the vertex of a quadratic function of the form
[tex]$$
T(t) = at^2 + bt + c
$$[/tex]
is given by
[tex]$$
t = -\frac{b}{2a}.
$$[/tex]
Substitute the values [tex]\( a = -0.01 \)[/tex] and [tex]\( b = 0.218 \)[/tex]:
[tex]$$
t = -\frac{0.218}{2(-0.01)} = \frac{0.218}{0.02} \approx 10.9.
$$[/tex]
Rounded to one decimal place, the maximum temperature is reached after [tex]\( 10.9 \)[/tex] hours.
Step 2. Find the maximum temperature.
Substitute [tex]\( t = 10.9 \)[/tex] into the function:
[tex]$$
T(10.9) = -0.01(10.9)^2 + 0.218(10.9) + 97.6.
$$[/tex]
Evaluating this expression gives approximately
[tex]$$
T(10.9) \approx 98.7881.
$$[/tex]
Rounded to one decimal place, the maximum temperature is [tex]\( 98.8^\circ \)[/tex]F.
Final Answers:
- The patient’s temperature reaches its maximum after [tex]\( 10.9 \)[/tex] hours.
- The patient’s maximum temperature is [tex]\( 98.8^\circ \)[/tex]F.
[tex]$$
T(t) = -0.01t^2 + 0.218t + 97.6,
$$[/tex]
where [tex]\( t \)[/tex] is in hours. Since the coefficient [tex]\( a = -0.01 \)[/tex] is negative, the quadratic function opens downward and therefore reaches a maximum at its vertex.
Step 1. Find the time at which the maximum occurs.
The time corresponding to the vertex of a quadratic function of the form
[tex]$$
T(t) = at^2 + bt + c
$$[/tex]
is given by
[tex]$$
t = -\frac{b}{2a}.
$$[/tex]
Substitute the values [tex]\( a = -0.01 \)[/tex] and [tex]\( b = 0.218 \)[/tex]:
[tex]$$
t = -\frac{0.218}{2(-0.01)} = \frac{0.218}{0.02} \approx 10.9.
$$[/tex]
Rounded to one decimal place, the maximum temperature is reached after [tex]\( 10.9 \)[/tex] hours.
Step 2. Find the maximum temperature.
Substitute [tex]\( t = 10.9 \)[/tex] into the function:
[tex]$$
T(10.9) = -0.01(10.9)^2 + 0.218(10.9) + 97.6.
$$[/tex]
Evaluating this expression gives approximately
[tex]$$
T(10.9) \approx 98.7881.
$$[/tex]
Rounded to one decimal place, the maximum temperature is [tex]\( 98.8^\circ \)[/tex]F.
Final Answers:
- The patient’s temperature reaches its maximum after [tex]\( 10.9 \)[/tex] hours.
- The patient’s maximum temperature is [tex]\( 98.8^\circ \)[/tex]F.
