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Chapter 1: Problem 24

Find the distance between the points. $$ (-2,1) \text { and }(10,6) $$

Short Answer

Expert verified
The distance between the points \((-2,1)\) and \((10,6)\) is \(13\) units.

Step by step solution

01

Identify the coordinates of the points

The two points are given as: Point A: \((-2, 1)\) Point B: \((10, 6)\)
02

Apply the distance formula

We will use the distance formula and substitute the coordinates of the points as follows: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ d = \sqrt{(10 - (-2))^2 + (6 - 1)^2} $$
03

Simplify the expression

Now, we will simplify the expression inside the square root: $$ d = \sqrt{(10 + 2)^2 + (6 - 1)^2} \\ d = \sqrt{12^2 + 5^2} $$
04

Calculate the distance

Finally, we will calculate the distance based on the simplified expression: $$ d = \sqrt{144 + 25} \\ d = \sqrt{169} $$ Since 169 has a square root equal to 13 (since \(13^2=169\)), the distance between the two points is: $$ d = 13 $$ So, the distance between the points \((-2, 1)\) and \((10, 6)\) is 13 units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Geometry
Coordinate geometry, also known as Cartesian geometry, is a mathematical system that connects algebra and geometry through the use of a coordinate plane. This concept allows us to describe geometric shapes and figures using numerical expressions and coordinates.
  • The coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
  • Every point on this plane is identified by a pair of numbers known as coordinates, written as \(x, y\).
  • The first number in this pair is the x-coordinate, which indicates the position along the x-axis, while the second number is the y-coordinate, showing the position along the y-axis.
By using these coordinates, we can precisely describe the location of points, the lengths of line segments, and even the equations of lines and curves. Coordinate geometry simplifies many problems in both math and science by providing a visual method to explore algebraic concepts.
Distance Calculation
The distance formula is a critical tool in coordinate geometry that helps calculate the straight-line distance between two points on the coordinate plane. Imagine trying to find the shortest path between two points. This formula provides an efficient method to determine that distance.
  • The distance formula is derived from the Pythagorean Theorem. It states: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
  • Here, \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two points.
  • The expression \(x_2 - x_1\) gives the difference in x-values, while \(y_2 - y_1\) shows the difference in y-values.
The formula calculates the hypotenuse of a right triangle formed by the horizontal and vertical distances between the points. By applying the formula, we can swiftly find the distance, simplifying many geometric problems.
Mathematical Problem Solving
Mathematical problem solving involves using logical reasoning to find a solution to a given problem. This process often requires a set of skills and techniques to break down complex problems into manageable steps.
  • First, identify what you need to solve. Clearly understanding the problem is crucial in devising an effective plan.
  • Next, determine which mathematical tools and formulas are relevant. For instance, knowing to use the distance formula when finding the distance between two points is essential.
  • Then, apply these formulas or techniques step-by-step as done in the calculation of the distance between the points \((-2, 1)\) and \(10, 6)\).
The value in practicing mathematical problem solving is not just in obtaining the correct answer but in developing a structured approach to addressing similar types of challenges in the future. This skill is not only essential in mathematics but also beneficial in everyday decision-making and problem-solving situations.

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Most popular questions from this chapter

The demand equation for the Schmidt- 3000 fax machine is \(3 x+p-1500=0\), where \(x\) is the quantity demanded per week and \(p\) is the unit price in dollars. The supply equation is \(2 x-3 p+1200=\) 0\. where \(x\) is the quantity the supplier will make available in the market each week when the unit price is \(p\) dollars. Find the equilibrium quantity and the equilibrium price for the fax machines.

The annual sales (in billions of dollars) of global positioning system (GPS) equipment from the year 2000 through 2006 follow \((x=0\) corresponds to the year 2000 ): $$ \begin{array}{lccccccc} \hline \text { Year, } \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Annual Sales, } \boldsymbol{y} & 7.9 & 9.6 & 11.5 & 13.3 & 15.2 & 16.0 & 18.8 \\ \hline \end{array} $$ a. Find an equation of the least-squares line for these data. b. Use the equation found in part (a) to estimate the annual sales of GPS equipment for 2008 , assuming that the trend continued.

If the slope of the line \(L_{1}\) is positive, then the slope of a line \(L_{2}\) perpendicular to \(L_{1}\) may be positive or negative.

Suppliers of a certain brand of digital voice recorders will make 10,000 available in the market if the unit price is \(\$ 45 .\) At a unit price of \(\$ 50,20,000\) units will be made available. Assuming that the relationship between the unit price and the quantity supplied is linear, derive the supply equation. Sketch the supply curve and determine the quantity suppliers will make available when the unit price is \(\$ 70\).

The lines with equations \(a x+b y+c_{1}=0\) and \(b x-a y+\) \(c_{2}=0\), where \(a \neq 0\) and \(b \neq 0\), are perpendicular to each other.
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