Challenges cover a variety of topics
You will need to use geometry and trigonometry to solve this challenge.
An ice cream shape is created when a cone is attached to a hemisphere. The angle at the bottom of the cone is 30 degrees and the volume of the structure is 500 millilitres. The challenge is to find the surface area of this structure.
For our final Fascinating Frequencies challenge, we will be using chemistry, physics, simulations, and orbits.
An experiment is held in space station orbiting earth from a geostationary orbit. Unfortunately, the scientists onboard forgot to properly secure one side of the space station. The side pointing prograde. Well, at least not retrograde, but their orbit is still messed up seriously. The reaction was between sodium and fluorine, with 19 kilograms fluorine and 23 kilograms sodium. The mass of the space station is 500 megagrams, including remaining fuel. How many grams of fuel must the scientists burn minimum to bring their rocket back to geostationary orbit? The scientists’ rocket fuses hydrogen and oxygen.
Algebra, trigonometry, and calculus are used in today’s challenge.
You are taking a streetcar through a city. Your destination is in the middle of a large park, and you are late. You can jump of the streetcar at any time while it is on the side of the park. Your destination is 1 kilometre from the road. The streetcar moves at 15 m/s, and you move at 5 when you run, which you will to try to be on time. You calculate when you will need to jump off the streetcar to arrive as soon as possible, and jump of exactly then. For how long do you run?
Geometry, trigonometry, and calculus are required for this challenge.
We left an unsolved puzzle at the end of Fascinating Frequencies 3. We want to find the time it takes for a stone to stop after hitting water. The stone has an initial velocity of 1 m/s, a mass of 1 kilogram, and radius 10cm. The force acting on it from gravity is 1kg*g down. The buoyant force acting on it is the weight of the water displaced by the stone. The water has density 0.9982 kg/m3. The friction from the water in any direction acts perpendicular to the surface, inwards, with a strength equal to the sine of the angle multiplied by the density of the water.
\vec F_G=1kg g\ down\\[8pt] \vec F_B=\frac{998.2kg gV_w}{m^3}\ up\\[8pt] \vec F_d=\frac{469.2kgA_f\vec V^2}{m^3}\ upYou will need to use physics for this Fascinating Frequencies challenge.
An interesting device can be created when you have a wheel of gears. Of course, this wheel will have an even number of gears. If the device is anchored and you spin one gear, the others will all also spin at the same frequencies. But if you anchor the gear you spin, the others can spin around it. Try building this device yourself, it’s very cool. Say you make one of these devices with six gears. The radius of each gear is 10 centimetres, and the centripetal force felt by the gear opposite to the anchored one you are spinning is 1 metre per second squared. The challenge is to find what frequency the gear is spinning.
A physics simulation is needed to solve this challenge.
Bob the Bungee jumper from Baffling Bungee is upset about his bungee jumping experience. If you attempt the problem, you will soon see that Bob’s bungee jump will take too long. Far too long. This is because C5H8 is very springy. In fact, Bob’s bungee jump would take about 2 months … if you were thinking of a mathematical model. You see, the diagram shows a cliff behind Bob. The rubber is so springy he would bash into it, cutting his jump short. But set that problem aside for now. This challenge does not concern those. In bob’s jumps, the ups and downs would gradually, very gradually, become smaller and smaller. But they would still occur at the same frequency. Can you guess today’s challenge? Calculate that frequency.
Combinatorics is needed to solve today’s challenge.
Shown in the diagram to the right is the board for a dice game. The rules are as follows: start at the point. On your turn, roll 2 six sided dice. Move forward thet many spaces allong the path. The path is marked by the arrow and has 24 squares. If you reach the black square and have more moves, move back to the dot. You do not continue moving; your turn ends immediately. The first player to end exactly on the black square wins. You play this game with a friend. The challenge is to find the average number of turns the game will take. After 1 player wins, the other does not continue. This is very important.
Some chemistry principles are required to solve this challenge.
In this Fascinating Frequencies challenge, we will cover fluid dynamics. A perfect homogeneous sphere with radius 10 cm and mass 1 kg hits the middle of a pond of pure water on earth. It’s velocity is 1 m/s downward. The temperature of the pond is a uniform 20° celsius. The pond will begin to ripple. There are no animals in the pond to interfere with this. The challenge is to calculate the frequency of the ripples.
Algebra is necessary to solve this challenge.
We left an unsolved system of equations in Fascinating Frequencies 2. In this challenge, you must solve it.
\vec v=\frac{\vec d\pi}{72000s}\\[8pt] \frac{4\pi\vec v\vec d}{sm}=\frac{Gm_1}{\vec d^2}\\[8pt] G=\frac{6.67408m^3}{10^{11}s^2kg}\\[8pt]In this challenge, you will need to use geometry, calculus, algebra, and angle theorems.
A roulette is a shape created by rolling things. If you roll circles, you have a trochoid. If you draw the path of a point on the circumference, you have a cycloid. And if you roll that circle inside another circle, you have a hypocycloid. Simple enough. Say you have a circle with radius 25 centimetres and a circle with radius 1 metre. When you rotate the smaller circle inside the bigger one, the path traveled by a point on the circumference is a hypocycloid. It will touch the bigger circle 4 times every rotation. At these points, the instantaneous velocity of the point is zero. The challenge is to find the instantaneous acceleration if the small circle completes 1 revolution inside the bigger one each second. This is different from rolling completely over the inside, which it will do every 4.