Let's break down the problem step by step:

Given Equation:

The equation provided for the dosage $p$ for a puppy of age $a$ (in weeks) is: $p = 0.0417D \times (a + 1)$ where $D$ is the recommended adult dosage (in ml).

The adult dosage $D$ is given as $100$ ml.

Substituting $D = 100$ ml:

Substitute $D = 100$ ml into the equation: $p = 0.0417 \times 100 \times (a + 1)$ Simplifying this, we get: $p = 4.17 \times (a + 1)$

Now let's solve the sub-questions:

a. Sketch the Graph for this Function

The function $p = 4.17 \times (a + 1)$ is a linear function in the form $y = mx + b$ where:

• $y$ is the response variable $p$ (dosage in ml),
• $x$ is the independent variable $a$ (age in weeks),
• $m = 4.17$ is the slope,
• $b = 4.17$ is the y-intercept (since when $a = 0$, $p = 4.17 \times 1 = 4.17$ ml).

b. Find the Slope of the Graph of $p$

From the function $p = 4.17(a + 1)$, the slope $m$ is the coefficient of $a$, which is $4.17$.

So, the slope $m$ of the graph is 4.17.

c. What is the Dosage for a 10-Week-Old Puppy?

To find the dosage $p$ for a 10-week-old puppy, substitute $a = 10$ into the equation: $p = 4.17 \times (10 + 1) = 4.17 \times 11$ $p = 45.87 \text{ ml}$

So, the dosage for a 10-week-old puppy is 45.87 ml.

d. At What Age Is a Dosage of 54.21 ml Used?

To find the age $a$ when the dosage is $54.21$ ml, we set $p = 54.21$ and solve for $a$: $54.21 = 4.17 \times (a + 1)$ Divide both sides by 4.17: $\frac{54.21}{4.17} = a + 1$ $a + 1 = 13$ $a = 12 \text{ weeks}$

So, a dosage of 54.21 ml is used at 12 weeks of age.

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Summary of Answers:

1. Graph: The graph is a straight line with a slope of 4.17 and a y-intercept at 4.17 ml.
2. Slope: The slope of the graph is 4.17.
3. Dosage for a 10-week-old puppy: 45.87 ml.
4. Age for 54.21 ml dosage: 12 weeks.

Would you like a detailed explanation of any part of this solution, or do you have any further questions?

Here are 5 related questions you might find interesting:

1. How would the graph change if the adult dosage $D$ was different?
2. What would the dosage be for a 5-week-old puppy?
3. What is the significance of the slope in the context of this problem?
4. How would you find the age of a puppy if the dosage required is known?
5. What does the y-intercept represent in the context of this function?

Tip: In linear functions, the slope determines the rate of change of the dependent variable (like dosage) with respect to the independent variable (like age).