Question

15. If the recommended adult dosage for a drug is \(D\) (in mg), then to determine the appropriate dosage \(c\) for a child of age \(a\), pharmacists use the equation \(c = 0.0417D\left( {a + 1} \right)\). Suppose the dosage for an adult is 200 mg.

(a) Find the slope of the graph of \(c\). What does it represent?

(b) What is the dosage for a newborn?

Short answer

(a) The slope of the equation \(c\) is 8.34. It represents the rate of change of the dosage for a child, in mg, with respect to a 1-year change in age.

(b) The dosage of a newborn is 8.34 mg.

Step-by-step solution

Describe the given information

It is given that the appropriate dosage of the drug is \(c = 0.0417D\left( {a + 1} \right)\), where \(D\) is the recommended dosage for an adult and \(a\) is the age of the child. It is also given that dosage of an adult is 200 mg.

Plug \(D = 200\) into \(c = 0.0417D\left( {a + 1} \right)\) and simplify

Substituting \(D = 200\) into \(c = 0.0417D\left( {a + 1} \right)\) and simplifying as shown below:

\(\begin{aligned}c &= 0.0417\left( {200} \right)\left( {a + 1} \right)\\ &= 8.34\left( {a + 1} \right){\rm{ }}\\ &= 8.34a + 8.34\end{aligned}\)

This equation can be compared with the slope-intercept form of the line \(y = mx + c\).

So, the slope of the equation \(c\) is 8.34. It represents the rate of change of the dosage for a child (in mg) with respect to 1-year change in age.

Plug  \(a = 0\) into \(c = 8.34a + 8.34\) and simplify

The age of a newborn is 0.

So, substitute\(a = 0\)into\(c = 8.34a + 8.34\) and simplifying as shown below:

\(\begin{aligned}c &= 8.34\left( 0 \right) + 8.34\\ &= 8.34\end{aligned}\)

So, the dosage of the newborn child is 8.34 mg.