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Chapter 1: Problem 25

Find the coordinates of the points that are 10 units away from the origin and have a \(y\) -coordinate equal to \(-6\).

Short Answer

Expert verified
The coordinates of the points that are 10 units away from the origin with a y-coordinate of -6 are (8, -6) and (-8, -6).

Step by step solution

01

Distance Formula

Recall the distance formula, which calculates the distance between two points (x1, y1) and (x2, y2) is: \(d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \) In this case, (x1, y1) is the origin (0,0) and (x2, y2) is the point we're trying to find, let's call it (x, -6). We are given that the distance between the origin and the point is 10 units, which means: \(d = 10\) Now we will use the distance formula with the given information.
02

Substitute Information into Distance Formula

Substitute (x1, y1) = (0,0), (x2, y2) = (x, -6), and d = 10 into the distance formula: \(10 = \sqrt{(x - 0)^2 + (-6 - 0)^2}\)
03

Solve for x

Now, simplify the equation and solve for x: \(10 = \sqrt{x^2 + 36} \) Square both sides to get rid of the square root: \(100 = x^2 + 36\) Now, subtract 36 from both sides: \(x^2 = 64\) Square root both sides to find x: \(x = \pm 8\)
04

Find Coordinates

We have two possible x-coordinates: 8 and -8. The y-coordinate is -6 as given. So, the coordinates of the points that are 10 units away from the origin with a y-coordinate of -6 are: (8, -6) and (-8, -6)

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Key Concepts

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

Distance Formula
The distance formula is a key tool in coordinate geometry. It helps us find the distance between two points on the coordinate plane. If you have two points, represented by \(x_1, y_1\) and \(x_2, y_2\), the distance \(d\) between them is calculated as follows:
  • Write the equation: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
  • This formula is derived from the Pythagorean theorem.
  • You subtract the x-coordinates and square the result, and then do the same for the y-coordinates.
  • Add these two squared terms together, then take the square root of the result.
To understand its application, consider the problem where the task is to find points that are 10 units away from the origin with a specific y-coordinate. By substituting the given values into the formula, you can calculate the unknown coordinates.
Coordinates of a Point
Coordinates of a point provide a specific location on the coordinate plane. A point is defined by an \(x\)-coordinate and a \(y\)-coordinate in the form (x, y). This system allows you to precisely locate points in a two-dimensional space:
  • The \(x\)-coordinate tells you how far to move horizontally.
  • The \(y\)-coordinate tells you how far to move vertically.
  • The origin has coordinates (0, 0), which is the central point where the x-axis and y-axis intersect.
In the example problem, we're given a y-coordinate of -6 and need to find the x-coordinates of points that are 10 units from the origin. Using both the fixed y-coordinate and the distance formula, we can solve for the potential x-coordinates. This illustrates how specific coordinates can shape our ability to pinpoint locations on the graph.
Mathematics Problem Solving
Successfully solving mathematics problems often requires a clear understanding of the problem and a structured approach to finding the solution. Here are some steps and tips:
  • Read and understand the problem carefully. Identify what is given and what needs to be found.
  • Translate the words into mathematical expressions or equations. For example, translating the problem's condition about distance into the distance formula.
  • Substitute provided values into the equations. In this exercise, this meant substituting known coordinates and distances into the distance formula.
  • Simplify the equation and solve for the unknowns. In our example, finding the x-coordinates involved solving a quadratic equation resulting from the distance formula.
  • Check your solution to ensure it meets all given conditions. Verify that the calculated points actually meet the distance requirement.
These steps help develop a methodical approach, ensuring not only that the solutions are correct, but that they are reached in an efficient and understandable manner. Problem-solving in mathematics is as much about using formulas correctly as it is about effectively reasoning through a problem's requirements.

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Most popular questions from this chapter

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.

Find an equation of the line that has slope \(m\) and \(y\) -intercept \(b\). $$ m=0 ; b=5 $$

The annual sales (in billions of dollars) of global positioning systems (GPS) equipment from 2000 through 2006 are shown in the following table \((x=0\) corresponds to 2000\()\) : $$ \begin{array}{lccccccc} \hline \text { Year, } \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Annual Sales, } \boldsymbol{y} & 7.9 & 9.6 & 11.5 & 13.3 & 15.2 & 17 & 18.8 \\ \hline \end{array} $$

For each pair of supply-and-demand equations, where \(x\) represents the quantity demanded in units of 1000 and \(p\) is the unit price in dollars, find the equilibrium quantity and the equilibrium price. $$ p=-2 x+22 \text { and } p=3 x+12 $$
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