Problem 64 - Algebra and Trigonometry with Analytic Geometry

Question

Pharmacological products must specify recommended dosages for adults and children. Two formulas for modification of adult dosage levels for young children are \(\begin{array}{lll} & \text { Cowling's rule: } & y=\frac{1}{24}(t+1) a \\\ \text { and } & \text { Friend's rule: } & y=\frac{2}{25} t a,\end{array}\) where \(a\) denotes adult dose (in milligrams) and \(t\) denotes the age of the child (in years). (a) If \(a=100\), graph the two linear equations on the same coordinate plane for \(0 \leq t \leq 12\). (b) For what age do the two formulas specify the same dosage?

Step-by-Step Solution

Answer

The two formulas specify the same dosage at approximately 1.09 years.

Step 1: Write the equations

First, substitute \(a = 100\) into both Cowling's rule and Friend's rule. Cowling's rule becomes \(y = \frac{1}{24}(t+1) \, 100\) which simplifies to \(y = \frac{100}{24}(t+1)\). Friend's rule becomes \(y = \frac{2}{25} \times t \times 100\) which simplifies to \(y = \frac{200}{25} t\) or \(y = 8t\).

Step 2: Graph the equations

Graph the equations \(y = \frac{100}{24} (t+1)\) and \(y = 8t\) over the domain \(0 \leq t \leq 12\). The first equation will start at \(y = \frac{100}{24}\) when \(t = 0\) and increase with a slope of \(\frac{100}{24}\). The second equation starts at the origin and increases with a slope of 8.

Step 3: Set equations equal to find intersection

To find the age \(t\) where the formulas specify the same dosage, set the equations equal: \(\frac{100}{24}(t+1) = 8t\). Solve for \(t\).

Step 4: Solve the equation

Expand and simplify the equation: \(\frac{100}{24}t + \frac{100}{24} = 8t\). Multiply everything by 24 to clear the denominator: \(100t + 100 = 192t\). Rearrange to get \(92t = 100\), then solve: \(t = \frac{100}{92} = \frac{25}{23}\) which is approximately \(1.09\) years.

Key Concepts

Linear Equations

When dealing with dosage calculations for children, two linear equations can help modify adult dosage levels. These are Cowling's rule and Friend's rule. Each formula represents a linear equation because they can be expressed in the form of \( y = mx + b \), where \( y \) is the unknown variable, \( x \) represents time or age, and \( m \) and \( b \) are constants. For example, Cowling's rule is expressed as \( y = \frac{100}{24}(t + 1) \) when substituting \( a = 100 \). On the other hand, Friend's rule becomes \( y = 8t \). The interesting part about these linear equations is how they exhibit a relationship between age and dosage with a linear increase in dosage as age increases. Understanding this concept is crucial as it simplifies the complexity of medication adjustments for varying ages.

Graphing Equations

Graphing linear equations is an essential step in visualizing relationships between variables, such as age and dosage in medication formulas. To graph the given linear equations derived from Cowling's and Friend's rules, you plot them on a coordinate plane where the horizontal axis represents age \( (t) \) and the vertical axis represents dosage \( (y) \). Start by identifying the y-intercept and slope for each equation.
For Cowling's rule, the y-intercept is \( \frac{100}{24} \) which indicates the dosage when \( t = 0 \). Its slope is \( \frac{100}{24} \), signaling how much the dosage increases with each year of age. Friend's rule starts at the origin with a slope of 8, meaning that for every year of age, the dosage increases by 8.
By graphing both equations on the same plot for ages 0 to 12, you can visually see where the dosages converge to determine when both rules suggest the same dosage.

Solving Equations

To find out when Cowling's rule and Friend's rule suggest the same dosage for children, we need to solve the equation \( \frac{100}{24}(t + 1) = 8t \). This entails finding the intersection point—the age where both equations yield the same result.
Start by expanding \( \frac{100}{24}(t + 1) \) to get \( \frac{100}{24}t + \frac{100}{24} \). Next, eliminate the fraction by multiplying throughout by 24, resulting in \( 100t + 100 = 192t \).
Rearranging gives \( 92t = 100 \), thus \( t = \frac{100}{92} = \frac{25}{23} \), meaning approximately 1.09 years. This calculation reveals that at around 1.09 years, both rules recommend the same dosage, thereby highlighting the practical implications of linear equations in medical dosage calculations.