Question bank: Rationalization of Denominators

In a practical engineering application, an angle of elevation from a point A to the top of a pole is measured as 45 degrees. A second point B, located at the base of the pole and in a straight line with point A and the top of the pole, forms an angle of 30 degrees with point A. This configuration is illustrated in the attached figure. If the distance between points A and B is 50 meters, calculate the height of the pole. (Hint: Consider the definition of the tangent of an angle and the relationship of the tangent of 45 degrees being equal to 1.)

In a geometry problem, a right triangle has one of the acute angles measuring 30 degrees and the hypotenuse has a length of 2. By defining the side opposite this angle as 'a' and the adjacent side as 'b', we know that the ratio between 'a' and 'b' will be the tangent of 30 degrees, which is equal to 1/√3. To express the length of the adjacent side in terms of the opposite side, we need to rationalize the denominator of the tangent ratio of 30 degrees. (1) Rationalize the denominator of the tangent ratio of 30 degrees and express 'b' in terms of 'a'. (2) What is the exact value of 'b' in this triangle?

In a sustainable construction project, an engineer needs to calculate the amount of material needed to cover the external surface of a structure. The area to be covered is represented by the expression (4 + 2√6) m^2. The coating material is sold in rolls with a width of 2 meters and a length of 20 meters. To optimize the use of the material, the engineer wants to cut the rolls in a way that the width of the material is an integer multiple of √6 meters. The first step of this process is to determine what should be the width, in meters, of each strip of material to be cut from the roll. To rationalize the calculation, the engineer needs to express the width of the strip in terms of √6 and integers. What is the width, in meters, of each strip of material that the engineer should cut from the roll? Justify your answer by rationalizing the denominator of the fraction that represents the width of the strip of material.

In a physics experiment, a block of mass 'm' is placed on an inclined plane with an angle of elevation 'θ' relative to the horizontal. To analyze the acting forces, it is convenient to decompose the weight of the block 'm' into two components: one parallel to the inclined plane and another perpendicular to it. The parallel component, denoted by 'P_//', is given by P_// = m * g * sin(θ), where 'g' is the acceleration due to gravity. The perpendicular component, denoted by 'P_⊥', is given by P_⊥ = m * g * cos(θ). To simplify the calculations, it is common to rationalize the sin(θ) and cos(θ) expressions that appear in the equations. Step 1: We start with P_// = m * g * sin(θ) and P_⊥ = m * g * cos(θ). Step 2: We rationalize the denominator of sin(θ) by taking a step that results in a new equation P_//' = m * g * a / b (where 'a' and 'b' are numbers) and keep P_⊥ as the original equation. Step 3: After rationalizing, we verify that P_//' = P_//. Based on this context and the indicated steps, answer the following questions: (1) Develop the reasoning of step 2 and show how to rationalize the denominator of sin(θ) to obtain the expression P_//'. (2) After rationalizing, prove that the obtained expression P_//' is equal to the original expression P_//, demonstrating that sin(θ) / √n = √n / k, where 'n' and 'k' are rational numbers.

During an experiment in physics, a student encountered the following situation: to calculate the intensity of the electric field at point P in a uniform field, he determined the ratio between the electric force acting on a test charge and the value of the test charge. This ratio is expressed by the formula E = F/q, where E is the intensity of the electric field, F is the electric force, and q is the test charge. When inserting the measured values for F and q, the student obtained an expression for E that included a square root in the denominator. However, to facilitate comparisons with other studies, he wants to rationalize the denominator, that is, to express the intensity of the electric field without the presence of the square root in the denominator. Considering that the original expression for E is E = \frac{2\sqrt{3}}{5\sqrt{2}}, rationalize the denominator and then explain how the new rationalized form helps in interpreting the value of the intensity of the electric field at point P.