Problem 33 - Precalculus with Calculus Previews

Question

Solve for the unknown. $$ P_{1}(x, 2,3), P_{2}(2,1,1) ; d\left(P_{1}, P_{2}\right)=\sqrt{21} $$

Step-by-Step Solution

Answer

The possible values for \(x\) are \(-2\) and \(6\).

Step 1: Recall the Distance Formula

The formula to calculate the distance between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in a three-dimensional space is given by: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\] We will use this formula to find the unknown \(x\).

Step 2: Substitute the Known Points

For our points \(P_1(x, 2, 3)\) and \(P_2(2, 1, 1)\), substitute into the distance formula:\[\sqrt{(2 - x)^2 + (1 - 2)^2 + (1 - 3)^2} = \sqrt{21}\]

Step 3: Simplify the Equation

First, simplify the squared terms:\[\sqrt{(2 - x)^2 + (-1)^2 + (-2)^2} = \sqrt{21}\]Which further simplifies to:\[\sqrt{(2 - x)^2 + 1 + 4} = \sqrt{21}\]So, the equation becomes:\[\sqrt{(2 - x)^2 + 5} = \sqrt{21}\]

Step 4: Eliminate the Square Root

To eliminate the square root, square both sides of the equation:\[(2 - x)^2 + 5 = 21\]Now, solve for the expression inside the parenthesis.

Step 5: Solve for \(x\)

Subtract 5 from both sides of the equation:\[(2 - x)^2 = 16\]Next, take the square root of both sides:\[2 - x = 4\]\[x = 2 - 4\]This simplifies to \(x = -2\).Since the square root can yield both positive and negative roots, consider the other possibility:\[2 - x = -4\]Solving for \(x\):\[x = 2 + 4\]Thus, \(x = 6\).

Step 6: Verify the Solution

Both \(x = -2\) and \(x = 6\) need to be verified. For \(x = -2\), the distances are:\[\sqrt{((-2) - 2)^2 + (2 - 1)^2 + (3 - 1)^2} = \sqrt{16 + 1 + 4} = \sqrt{21}\].For \(x = 6\):\[\sqrt{(6 - 2)^2 + (2 - 1)^2 + (3 - 1)^2} = \sqrt{16 + 1 + 4} = \sqrt{21}\].Both solutions satisfy the distance condition.

Key Concepts

Three-Dimensional Space

Understanding three-dimensional space is crucial when working with points in geometry. In three-dimensional space, each point is determined by three coordinates:

• The x-coordinate which represents the position along the horizontal axis (left or right).
• The y-coordinate showing the position on the vertical axis (up or down).
• The z-coordinate indicating the depth or third dimension, which represents moving forward or backward.

This spatial understanding allows us to calculate distances and solve complex equations involving these three axes.
To visualize this, imagine navigating through a building:

• Going up or down floors represents the z-axis.
• Moving along a hallway represents changes in x and y coordinates.

Our exercise involves finding the distance between two given points, which are expressed in these three coordinates. Knowing how to navigate and plot points in this three-dimensional space is the first step to applying the distance formula effectively.

Solving for Unknowns

When solving problems in math, especially in geometry and algebra, we often need to find unknown variables. In our exercise, the unknown is the variable \(x\) in the point

• \(P_1(x, 2, 3)\)

Here, the goal is to use the given conditions to find all possible values of \(x\).
We use the distance formula in three-dimensional space to equate given known values, which allows us to solve for this unknown.

• First, substitute the coordinates from both points into the distance formula.
• Plug in the distance value, which in this case is \(\sqrt{21}\), and manipulate the equation to isolate \(x\).

Finding unknown variables involves arranging and simplifying expressions so that you can isolate the variable in question, which leads us to our next core concept.

Equation Simplification

Equation simplification is a skill that involves making an equation easier to work with. This is essential in solving for unknowns. The given exercise required simplification at various stages.
Starting with the equation

• \(\sqrt{(2 - x)^2 + 5} = \sqrt{21}\),

we need to simplify to solve for \(x\).
Firstly, simplification involves eliminating complex elements by performing operations such as:

• Squaring both sides to eliminate the square root, making it simpler: \((2 - x)^2 + 5 = 21\).
• Subtracting 5 from both sides to isolate squared terms: \((2 - x)^2 = 16\).

These steps not only simplify calculations, but they also make it easier to find meaningful solutions.
The process illustrates how breaking down a complex equation helps in uncovering potential solutions or unknown elements involved.