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Showing posts from December, 2023

Statistical Showdown: The Mysterious Case of Missing Data

A group of detectives are investigating a series of mysterious disappearances in the town of Numeralia. They've gathered data on the ages of the 30 missing people, which are as follows: 15, 22, 19, 17, 34, 21, 18, 20, 16, 22, 25, 23, 18, 17, 19, 21, 20, 16, 15, 24, 20, 23, 25, 22, 17, 19, 21, 16, 15, 18. As a budding statistician, your task is to help the detectives by providing the following statistical insights: Calculate the mean, median, and mode of the ages. Determine the range of the data. Comment on what these statistics can tell us about the age group of the missing people. Provide your calculations and insights to assist the detectives in understanding if there's a pattern in the ages of the missing individuals.

The Parabolic Pathfinder

A parabolic path is modeled by the quadratic equation � = − 2 � 2 + 12 � − 10 y = − 2 x 2 + 12 x − 10 , where � y represents the height in meters above ground and � x represents the horizontal distance in meters from the starting point. At what horizontal distances � x is the path exactly 6 meters above the ground? What is the maximum height reached by the path and at what horizontal distance does this occur? Provide your answers in terms of meters and round to two decimal places if necessary.

The Paradox of the Infinite Hotel

Consider a hotel with an infinite number of rooms, each numbered consecutively from 1 onwards. The hotel is fully occupied, but a new guest arrives. To accommodate this guest, the manager decides to move the guest in Room 1 to Room 2, the guest in Room 2 to Room 3, and so on, shifting each existing guest to the room number that is one greater than their current room number. After this rearrangement, in which room will the new guest be accommodated?

The Mystery of the Missing Angle

  In a triangle, one of the angles is twice as large as the smallest angle, while the third angle is 20 degrees less than the sum of the other two angles. Can you determine the measures of all three angles in this mysterious triangle?

The Secret of the Prime Pyramid

  An ancient pyramid has a numerical puzzle engraved on its entrance. The puzzle states that there are three prime numbers, each less than 20, that form a unique set. The sum of these three prime numbers is equal to the square of the smallest prime number in the set. What are these three prime numbers? Remember, a prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

The Labyrinth of Logarithmic Mysteries

  In a mythical labyrinth, a treasure chest is locked behind a door with a log-based code. The code is the solution to the following logarithmic equation: log ⁡ 2 ( � ) + log ⁡ 2 ( � − 4 ) = 3 lo g 2 ​ ( x ) + lo g 2 ​ ( x − 4 ) = 3 . To unlock the chest and claim the treasure, you must find the value of � x that satisfies this equation. Remember, logarithmic properties might be helpful in solving this equation.

The Riddle of the Rotating Satellite

A communication satellite orbits the Earth in a circular path. At a certain time, its shadow falls on a particular point on the Earth's surface. After 3 hours, the Earth has rotated 45 degrees, and the satellite has also completed a part of its orbit. If the satellite's shadow is now 1000 km away from the original point, calculate the radius of the satellite's orbit, assuming it remains constant and the Earth is a perfect sphere with a radius of 6400 km. Use the formula for the arc length in a circle, S=rθ, where S is the arc length, r is the radius, and θ is the angle in radians.