Question bank: Rationalization of Denominators

In a physics project, a student observes the movement of an object following a parabolic trajectory. The equation describing the height 'h' of the object at time 't' is given by h = -5t^2 + 20t + 10, where 'h' is in meters and 't' in seconds. To analyze the movement, the student decides to calculate the velocity of the object at a specific instant 't'. By deriving the height function with respect to time, he obtained the velocity function, v(t) = -10t + 20. Now, he wants to calculate the velocity of the object at the instant t = 2 seconds. Based on this data:

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 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.)

During the construction of an amusement park, an architect needed to calculate the height of a new extreme ride to ensure the safety of the users. The height of the free fall was designed to create a sensation of extreme speed, with the tracks described by a parabola in the air. To ensure the integrity of the structure, the architect needs to calculate the ratio between the velocity of a cart at the highest point of the parabola and the gravitational force pulling it down, represented by the weight of the cart. This ratio is directly influenced by the angle formed by the parabola in relation to the ground, which, in turn, depends on the height of the highest point. If the height of the highest point is represented by √10 meters, and the desired ratio is expressed as a fraction where the numerator is the square root of g (√g) and the denominator is the mass of the cart, in kilograms, rationalize the expression that represents the velocity of the cart at the highest point, in order to correctly identify the gravitational force acting on it.

In a practical application of mathematical concepts, such as physics, we often come across expressions that include square roots in the denominators. Rationalizing the denominators of the resulting fractions is a fundamental step to simplify calculations and interpret the results. Consider the following expression that arises in a kinematics problem: \[ \frac{5}{\sqrt{3}} + \frac{4}{2\sqrt{5}}. \] Step 1: Rationalize the denominator of \(\frac{5}{\sqrt{3}}\) and \(\frac{4}{2\sqrt{5}}\) separately, applying the concept of multiplying by a convenient form of 1 to not alter the value of the expression. Step 2: After rationalizing both terms, simplify the resulting expression by combining the like terms.