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Answer :
Sure! Let's break down the problem step by step.
### Part (a)
1. Interpreting [tex]$f(170) = 124$[/tex]
This statement indicates that when a patient weighs 170 pounds, the appropriate dosage of the painkiller is 124 milligrams.
2. Interpreting [tex]$f'(170) = 5$[/tex]
This statement tells us how the dosage changes as the patient's weight changes. Specifically, it says that for each additional pound the patient's weight increases from 170 pounds, the dosage should be increased by 5 milligrams. It's like a rate of change or a slope.
### Part (b)
1. Estimating [tex]$f(177)$[/tex]
We need to estimate the dosage for a patient weighing 177 pounds. Since we know [tex]$f(170) = 124$[/tex] mg for a 170-pound patient and that the dosage should increase by 5 mg for each additional pound according to [tex]$f'(170) = 5$[/tex], we can use this information.
- Calculate the weight difference: The patient weighs 177 pounds, which is 7 pounds more than 170 pounds (177 - 170 = 7).
- Calculate the dosage increase: With a rate of 5 mg per additional pound, the dosage should increase by [tex]\(5 \cdot 7 = 35\)[/tex] mg.
- Calculate the estimated dosage for 177 pounds:
[tex]\[
f(177) = f(170) + 35 = 124 + 35 = 159 \text{ mg}
\][/tex]
So, for a patient weighing 177 pounds, the estimated dosage would be 159 milligrams.
### Part (a)
1. Interpreting [tex]$f(170) = 124$[/tex]
This statement indicates that when a patient weighs 170 pounds, the appropriate dosage of the painkiller is 124 milligrams.
2. Interpreting [tex]$f'(170) = 5$[/tex]
This statement tells us how the dosage changes as the patient's weight changes. Specifically, it says that for each additional pound the patient's weight increases from 170 pounds, the dosage should be increased by 5 milligrams. It's like a rate of change or a slope.
### Part (b)
1. Estimating [tex]$f(177)$[/tex]
We need to estimate the dosage for a patient weighing 177 pounds. Since we know [tex]$f(170) = 124$[/tex] mg for a 170-pound patient and that the dosage should increase by 5 mg for each additional pound according to [tex]$f'(170) = 5$[/tex], we can use this information.
- Calculate the weight difference: The patient weighs 177 pounds, which is 7 pounds more than 170 pounds (177 - 170 = 7).
- Calculate the dosage increase: With a rate of 5 mg per additional pound, the dosage should increase by [tex]\(5 \cdot 7 = 35\)[/tex] mg.
- Calculate the estimated dosage for 177 pounds:
[tex]\[
f(177) = f(170) + 35 = 124 + 35 = 159 \text{ mg}
\][/tex]
So, for a patient weighing 177 pounds, the estimated dosage would be 159 milligrams.
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