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Chapter 2: Problem 100

Exercises \(100-102\) will help you prepare for the material covered in the next section. Let \(\quad\left(x_{1}, y_{1}\right)=(7,2) \quad\) and \(\quad\left(x_{2}, y_{2}\right)=(1,-1) . \quad\) Find \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} .\) Express the answer in simplified radical form.

Short Answer

Expert verified
The distance between the two points is \(3\sqrt{5}\) units

Step by step solution

01

Identify the Given Points

Given two points in the form (x, y) - point 1 is (7, 2) and point 2 is (1, -1)
02

Implement the Distance Formula

The formula for the distance between two points (x1, y1) and (x2, y2) is \(\sqrt{(x2 - x1)^2 + (y2 - y1)^2}\)
03

Substitute the given points into the formula

Substitute the coordinates of the given points into the distance formula: \(\sqrt{(1 - 7)^2 + (-1 - 2)^2}\)
04

Perform the calculations

Calculate the individual squared differences, then their sum, and lastly take the square root of that sum to get the distance: \(\sqrt{(-6)^2 + (-3)^2} = \sqrt{36 + 9}= \sqrt{45}\)
05

Simplify the radical

To simplify \(\sqrt{45}\), find the prime factorization of 45 which is \(3 * 3 * 5\), then take out in pairs. This gives \(3\sqrt{5}\)

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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 analytic geometry, merges algebra and geometry using a coordinate system to solve geometry problems. It allows us to calculate distances, angles, and other properties, by placing geometric figures within a coordinate system and using formulas, especially useful when working with planes.
  • In coordinate geometry, every point is defined by a set of coordinates \((x, y)\).
  • You can find geometric relationships like midpoints, slopes, and distances between points algebraically.
By assigning coordinates to geometric shapes, complex geometric problems become simple arithmetic tasks. In this exercise, we are dealing with a standard application of coordinate geometry: calculating the distance between two points.
Simplified Radical Form
When dealing with expressions that involve square roots, it鈥檚 often best to simplify them into their most reduced or simplified form. This process ensures the expression is in a format that is easiest to read or use in further calculations.
  • Simplifying a square root involves factoring the number under the square root into its prime factors.
  • You then group the factors into pairs, where each pair can become a factor outside the square root.
For instance, given \(\sqrt{45}\), we find its prime factors: \(45 = 3 \times 3 \times 5\). We can simplify this to \(3\sqrt{5}\), because the pair of 3's come out of the square root as a single 3. This technique is critical in making complex calculations more manageable and results more precise.
Distance Between Points
One of the primary uses of coordinate geometry is finding the distance between two points on a plane. The distance formula is derived from the Pythagorean Theorem, and it allows you to calculate this distance using algebra.
  • The formula to determine the distance \(d\) between points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  • This expression calculates the horizontal and vertical distances between the points and then uses these to find the total distance by applying the essence of the Pythagorean Theorem.
In our exercise, substituting the points \((7, 2)\) and \((1, -1)\) into the formula results in \(\sqrt{(-6)^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45}\). This method provides an efficient way to determine the exact distance, which is crucial for accurate measurements in various fields like physics, navigation, and computer graphics.

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Most popular questions from this chapter

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ (x+2)^{2}+(y+2)^{2}=4 $$

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{aligned} (x-2)^{2}+(y+3)^{2} &=4 \\ y &=x-3 \end{aligned} $$

Exercises \(129-131\) will help you prepare for the material covered in the next section. The function \(C(t)=20+0.40(t-60)\) describes the monthly cost, \(C(t),\) in dollars, for a cellphone plan for \(t\) calling minutes, where \(t>60 .\) Find and interpret \(C(100)\)

The bar graph shows your chances of surviving to various ages once you reach 60 The functions $$ \begin{aligned} f(x) &=-2.9 x+286 \\ \text { and } g(x) &=0.01 x^{2}-4.9 x+370 \end{aligned} $$ model the chance, as a percent, that a 60 -year-old will survive to age \(x .\) Use this information to solve Exercises \(101-102\) a. Find and interpret \(f(90)\) b. Find and interpret \(g(90)\) c. Which function serves as a better model for the chance of surviving to age \(90 ?\)

In Exercises \(105-108,\) you will be developing functions that model given conditions. You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. Write the total time, \(T,\) in hours, devoted to your outgoing and return trips as a function of your rate on the outgoing trip, \(x .\) Then find and interpret \(T(30) .\) Hint: Time traveled \(=\frac{\text { Distance traveled }}{\text { Rate of travel }}\)
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