Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Approximate each expression to the nearest hundredth. $$\sqrt{(4-6)^{2}+(7+1)^{2}}$$

Step-by-Step Solution

Answer

The approximate value is 8.25.

Step 1: Simplify Inside the Parentheses

First, simplify the expressions inside the parentheses. Calculate \((4 - 6)\) and \((7 + 1)\). We get:\[4 - 6 = -2 \quad \text{and} \quad 7 + 1 = 8\]

Step 2: Square Each Value

Next, square the results from Step 1. Calculate \((-2)^2\) and \(8^2\). We get:\[(-2)^2 = 4 \quad \text{and} \quad 8^2 = 64\]

Step 3: Add the Squared Values

After squaring, add these results together. Calculate:\[4 + 64 = 68\]

Step 4: Take the Square Root

Finally, we take the square root of the result from Step 3 to find the approximate value:\[\sqrt{68} \approx 8.2462\]

Step 5: Round to the Nearest Hundredth

Round \(8.2462\) to the nearest hundredth. So, we have:\[8.2462 \approx 8.25\]

Key Concepts

Square Root

The square root is a fundamental mathematical concept used in various areas, including finding distances in coordinate geometry. When we say square root, we're talking about a number that, when multiplied by itself, gives a specific value. In this exercise, we are finding the square root of the sum of squares of certain numbers.

The notation \( \sqrt{68} \) signifies the operation of determining what number, multiplied by itself, equals 68. This is a crucial step in problems involving the distance formula, where we seek to find the length of a segment in a coordinate plane.

Often, square roots result in irrational numbers, characterized by non-repeating and non-terminating decimal expansions. In practical exercises, such as this one, it's common to use approximations, especially up to given decimal places, like hundredths. Here, \( \sqrt{68} \approx 8.2462 \), which rounds to \( 8.25 \) when rounded to the nearest hundredth. Mastery of rounding is essential for accuracy in mathematical results.

Pythagorean Theorem

The Pythagorean Theorem is a crucial principle in geometry that explains the relationship between the lengths of sides in a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In symbolic terms, if \( a \) and \( b \) are the legs of a right triangle, and \( c \) is the hypotenuse, then the theorem is given by \( a^2 + b^2 = c^2 \). This theorem forms the foundation of the distance formula used in coordinate geometry.

The given exercise essentially applies the Pythagorean Theorem to calculate a distance in a coordinate plane. By treating the differences in x and y coordinates as the legs of a right triangle, we can use this theorem to find the straight-line distance between two points — a direct application of its principles.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. One of its main applications is to find the distance between two points in a plane.

• The points are defined by their coordinates, often marked as (x, y).
• The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the formula: \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula is derived from the Pythagorean Theorem, whereby the horizontal and vertical distances between the points form the legs of a right triangle.

In the original problem, we are effectively using this formula to calculate the distance between two points. This practical use of coordinate geometry is instrumental in various fields such as physics and engineering. By understanding the coordinate plane and calculations like these, students gain a deeper insight into how different positions relate to each other in space.