Question

If the recommended adult dosage for a drug is\({\bf{D}}\left( {{\bf{in}}\;{\bf{mg}}} \right)\), then to determine the appropriate dosage\({\bf{c}}\)for a child of age\({\bf{a}}\), pharmacists use the equation\({\bf{c = 0}}{\bf{.0417D(a + 1)}}\). Suppose the dosage for an adult is 200 mg.

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

(b) What is the dosage for a newborn?

Short answer

(a). The slope of the graph of\(c\)is 8.34.It represents that the dosage will increase by 8.34 mg if age increases by 1 year.

(b). The dosages for a newborn are\(8.34\;mg\).

Step-by-step solution

Given data

Suppose that, the appropriate dosage\(c\)for a child of age\(a\), pharmacists use the equation\(c = 0.0417D\left( {a + 1} \right)\)

Here D represents the recommended adult dosage for a drug in mg.

If the dosage for an adult is 200mg, then D=200 mg.

:(a) Determine the slope of the graph of\({\bf{c}}\)and what does it represent

It needs to determine the slope of the graph of c.

Substitute\(D = 200\)in the given equation.

\(\begin{array}{l}\\\\\end{array}\)\(\begin{array}{c}c = 0.0417D\left( {a + 1} \right)\\c = 0.0417\left( {200} \right)\left( {a + 1} \right)\\c = 8.34\left( {a + 1} \right)\\c = 8.34\left( a \right) + 8.34\end{array}\)

On comparing with the equation\(y = mx + b\), we get\(m = 8.34\).

Therefore, the slope of the graph of\(c\)is 8.34.

The graph is given below

The below figure shows the graph of the linear function\(c\left( a \right) = 8.34a + 8.34\).

Thus,the slope of the graph\(c\left( a \right) = 8.34a + 8.34\)represents the rate of change of\(c\left( a \right)\)with respect to\(a\). It represents the increases in dosage in mg as age increases. Since the slope is 8.34, therefore the dosage will increase by 8.34 mg if age increases by 1.

(b) Determine the dosage for a newborn

For a newborn, we use\(a = 0\)and calculate\(c\)

\(\begin{array}{c}c = 8.34\left( a \right) + 8.34\\ = 8.34\left( 0 \right) + 8.34\\ = 8.34\end{array}\)

So, the dosage for a newborn is mg