Question

a formula that approximates the dosage of a drug prescribed for children is shown below. in the formula, a = the childs age, in years, d = an adult dosage, and c = the proper childs dosage. the formula applies for ages 2 through 13, inclusive. use youngs rule to find the difference in a childs dosage for a 9 - year - old child and a 6 - year - old child. express the answer as a single rational expression in terms of d. then describe what your answer means in terms of the variables in the model.
youngs rule: c = \frac{da}{a + 12}
the difference in dosage is \square. (simplify your answer. use integers or fractions for any numbers in the expression.)
describe what this result means in terms of the variables in the model.
this is the difference in dosage for a 9 - year - old child and a 6 - year - old child, as a \frac{\text{fraction of the difference in age}}{\text{multiple of the sum of the ages}} \frac{\text{multiple of the sum of adult dosages}}{\text{fraction of an adult dosage}}.

Explanation:

Step1: Calculate the dosage for 9 - year - old child

Substitute \(A = 9\) into Young's rule \(C=\frac{DA}{A + 12}\). So \(C_1=\frac{9D}{9+12}=\frac{9D}{21}=\frac{3D}{7}\)

Step2: Calculate the dosage for 6 - year - old child

Substitute \(A = 6\) into Young's rule \(C=\frac{DA}{A + 12}\). So \(C_2=\frac{6D}{6 + 12}=\frac{6D}{18}=\frac{D}{3}\)

Step3: Find the difference in dosages

\(C_1-C_2=\frac{3D}{7}-\frac{D}{3}\). Get a common denominator, which is \(21\). Then \(\frac{3D}{7}-\frac{D}{3}=\frac{9D}{21}-\frac{7D}{21}=\frac{9D - 7D}{21}=\frac{2D}{21}\)

Answer:

\(\frac{2D}{21}\). This means that the difference in dosage between a 9 - year - old and a 6 - year - old child is \(\frac{2}{21}\) of the adult dosage \(D\).