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Chapter 1: Problem 36

A certain corner of a room is selected as the origin of a rectangular coordinate system. If a fly is crawling on an adjacent wall at a point having coordinates (2.0, \(1.0)\), where the units are meters, what is the distance of the fly from the corner of the room?

Short Answer

Expert verified
The distance of the fly from the corner of the room or the origin of the coordinate system is \(\sqrt{5.0}\) meters.

Step by step solution

01

Identify the Distance Formula

The formula to calculate distance in a two-dimensional space follows from the Pythagorean theorem, and it is: \[d = \sqrt{x^2 + y^2}\]. Here, d is the distance, x is the x-coordinate and y is the y-coordinate.
02

Substitute the Known Values into the Distance Formula

Our known values from the problem are x = 2.0 and y = 1.0. Replacing these values into the equation, we get: \[d = \sqrt{(2.0)^2 + (1.0)^2}\].
03

Calculate the Distance

Solving the equation: \[d = \sqrt{(4.0) + (1.0)} = \sqrt{5.0}\].

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Key Concepts

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

Rectangular Coordinate System
The rectangular coordinate system, also known as the Cartesian coordinate system, is a foundational tool in mathematics for representing points in two-dimensional space. Imagine a flat surface with two intersecting lines, one horizontal and one vertical. These lines are referred to as axes. The horizontal axis is called the x-axis, while the vertical one is the y-axis.

Each point on this surface can be identified by an ordered pair of numbers, written as (x, y), where 'x' represents the position along the horizontal axis and 'y' represents the position along the vertical axis. The point at which the axes intersect is called the origin, usually denoted as (0, 0).

In our exercise, the corner of the room is the origin of our coordinate system, and the fly's position at (2.0, 1.0) tells us that it is located 2 meters along the x-axis (horizontally) and 1 meter along the y-axis (vertically) from the origin.
Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. The theorem states that in such a triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

This relationship is expressed algebraically as \(a^2 + b^2 = c^2\), where 'a' and 'b' are the lengths of the legs of the triangle, and 'c' is the length of the hypotenuse.

In our exercise, we use this theorem to find the distance from the origin to the fly, which forms a right-angled triangle with the axes. The coordinates (2.0, 1.0) represent the lengths of the legs (a and b), and our goal is to find the hypotenuse (c), which is the straight-line distance from the origin to the fly.
Two-Dimensional Space
Two-dimensional space, or 2D space, refers to a geometric setting in which only two dimensions are considered for any point or object within it. Length and width are the dimensions of this plane, which can be represented on a flat surface, such as paper or a computer screen. Positions in two-dimensional space can be described using two coordinates in a rectangular coordinate system.

By understanding two-dimensional space, we can visualize and solve many real-world problems that are confined to flat surfaces. The distance measuring exercise involves determining the straight-line path within a plane from the origin to a specific point (the fly's position), illustrating a practical application of concepts in two-dimensional space.

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Most popular questions from this chapter

Sphere 1 has surface area \(A_{1}\) and volume \(V_{1}\), and sphere 2 has surface area \(A_{2}\) and volume \(V_{2}\). If the radius of sphere 2 is double the radius of sphere 1 , what is the ratio of (a) the areas, \(A_{2} / A_{1}\) and (b) the volumes, \(V_{2} / V_{1}\) ?

A right triangle has a hypotenuse of length \(3.00 \mathrm{~m}\), and one of its angles is \(30.0^{\circ}\). What are the lengths of (a) the side opposite the \(30.0^{\circ}\) angle and (b) the side adjacent to the \(30.0^{\circ}\) angle?

A high fountain of water is located at the center of a circular pool as shown in Figure P1.43. Not wishing to get his feet wet, a student walks around the pool and measures its circumference to be \(15.0 \mathrm{~m}\). Next, the student stands at the edge of the pool and uses a protractor to gauge the angle of elevation at the bottom of the fountain to be \(55.0^{\circ}\). How high is the fountain?

Kinetic energy \(K E\) (Chapter 5 ) has dimensions \(\mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}\). It can be written in terms of the momentum \(p\) (Chapter 6 ) and mass \(m\) as $$ K E=\frac{p^{2}}{2 m} $$ (a) Determine the proper units for momentum using dimensional analysis. (b) Refer to Problem 5. Given the units of force, write a simple equation relating a constant force \(F\) exerted on an object, an interval of time \(t\) during which the force is applied, and the resulting momentum of the object, \(p\).

A fathom is a unit of length, usually reserved for measuring the depth of water. A fathom is approximately \(6 \mathrm{ft}\) in length. Take the distance from Earth to the Moon to be 250000 miles, and use the given approximation to find the distance in fathoms.
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