Let's break down the problem to provide a detailed solution.

The question asks to calculate the distance between points $C$ and $D$ (which you found as $C(4, 27)$ and $D(-1, 42)$) and express the distance in the form $\frac{\sqrt{p}}{q}$, where $p$ and $q$ are integers.

Step-by-Step Solution

The distance formula between two points $C(x_1, y_1)$ and $D(x_2, y_2)$ is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

For the points $C(4, 27)$ and $D(-1, 42)$, substitute the coordinates into the formula:

$d = \sqrt{((-1) - 4)^2 + (42 - 27)^2}$

Calculating the differences:

$d = \sqrt{(-5)^2 + (15)^2}$

$d = \sqrt{25 + 225}$

$d = \sqrt{250}$

Now, simplify $\sqrt{250}$:

$d = \frac{\sqrt{250}}{1} = \frac{\sqrt{25 \times 10}}{1} = \frac{5\sqrt{10}}{1}$

Thus, the distance between points $C$ and $D$ is $\frac{5\sqrt{10}}{1}$.

Final Answer:

The distance $CD$ is $\frac{5\sqrt{10}}{1}$, with $p = 250$ and $q = 1$.

Do you have any questions or would you like further details?

Here are 5 related questions for further exploration:

1. How would the result change if one of the coordinates was negative?
2. Can you generalize the distance formula for three-dimensional space?
3. What does the distance between points represent geometrically?
4. How can we verify the solution using graph plotting?
5. How would you calculate the midpoint of points $C$ and $D$?

Tip: When simplifying square roots, try to factor out perfect squares to make calculations easier!