Cowling's Rule is a method for calculating pediatric drug dosages. If a denotes the adult dosage (in milligrams) and if t is the child's age (in years), then the child's dosage is given by
D(t) = (t + 1/24)a
(a) Show that D is a linear function of t.
Hint: Think of D(t) as having the form
D(t) = mt + b.
What are the slope m and the y-intercept b?
m = ______
b = ______
m = a, b = a/24
To show that D is a linear function of t, we need to express D(t) in the form D(t) = mt + b, where m is the slope and b is the y-intercept.
Given that D(t) = (t + 1/24)a, let's rewrite it in the form D(t) = mt + b:
D(t) = (1/24)a + ta
= (1/24)a + (24/24)ta
= (1/24)a + (24ta/24)
Now, we can see that the coefficient of t is 24a/24, which simplifies to a. Hence, the slope of D(t) is m = a.
The y-intercept, b, is the value of D when t = 0. When t = 0, we have:
D(0) = (0 + 1/24)a
= a/24
Therefore, the y-intercept, b, is b = a/24.
Hence, the linear function D(t) is D(t) = at/24 + a/24.
The slope m is a.
The y-intercept b is a/24.
To show that D(t) is a linear function of t, we need to write it in the form D(t) = mt + b.
Given that D(t) = (t + 1/24)a, we can rewrite it as:
D(t) = (a/24)t + a
Comparing this equation with the general form of a linear function, D(t) = mt + b, we can identify the slope m and the y-intercept b.
The slope, m, is the coefficient of t in the equation, which is a/24.
Therefore, m = a/24.
The y-intercept, b, is the constant term in the equation, which is a.
Therefore, b = a.
So, the slope (m) of the function D(t) is a/24, and the y-intercept (b) is a.