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# Monthly Archives: December 2023

## Total Number of Triangles Puzzle

How many triangles are there in the figure below?

Solution to this Geometry Triangle Puzzle problem is given in the video below!

## Total Number of Rectangles Puzzle

How many rectangles are there in the figure below?

Solution to this Geometry Rectangle Puzzle is given in the video below!

## Total Number of Upright SQUARES in a Square

How many upright Squares are there in the following square?

The key is to understand vertices of any given upright square and a diagonal line of the largest square to see how many translations of that given upright square you can have. As such, there is a summation-derived FORMULA you can actually use for any largest size square to find the total number of possible upright squares contained within it!

Solution to this Geometry Square Puzzle is given in the video below!

## Total Number of Upright RECTANGLES in a Square (or Rectangle)

How many upright Rectangles are there in the following square?

The key is to understand vertices of any given upright rectangle to see how many unique translations of that rectangle are possible. As such, there is actually a summation-derived FORMULA you can utilize for the largest square or rectangle of any size to find the total number of possible upright rectangles within it!

Solution to this Geometry Rectangle Puzzle is given in the video below!

## Total Number of Upright SQUARES in a Rectangle

How many upright Squares are there in the following rectangle?

The key is to understand vertices of any given upright square to see how many translations of that given square you can have. As such, there is a summation-related FORMULA you can actually derive for any size rectangle to find the total number of possible upright squares contained within it!

Solution to this Square Rectangle Geometry Puzzle is given in the video below!

## Total Number of Upright & Tilted SQUARES in a Square

How many upright and tilted Squares are there in the following square?

The key is to use vertices of any given largest square to see how many translations of a given smaller square you can have. As such, there is a summation-derived FORMULA you can actually use for any size square to find the total number of possible upright and tilted squares contained within it!

Solution to this Square Geometry Puzzle example is given in the video below!

## Total Number of Upright & Tilted SQUARES in a Rectangle

How many upright and tilted Squares are there in the following rectangle?

The key is to use vertices of any given rectangle to see how many translations of a given square you can have. As such, there is a summation-derived FORMULA you can actually use for any size rectangle to find the total number of possible upright and tilted squares contained within it!

Solution to this Rectangle Geometry Puzzle example problem is given in the video below!

## Two given functions are Inverses of each other? Example problem

Determine whether the given two functions are Inverses.

a. f(x) = x – 5; g(x) = x + 5

b.

c.

d.

Solution to this Inverse Function example practice problem is provided in the video below!

## Ferris Wheel trigonometry word problem

The Ferris wheel at Navy Pier has a diameter of 140 feet. It stands 10 feet off the ground. The wheel has 40 gondolas that seat six passengers each. It takes about 6 minutes for the Navi Pier Ferris wheel to complete one rotation.

Draw a diagram of the Navy Pier Ferris wheel and the boarding platform. Fill in the necessary information. Sketch the graph. Write a cosine equation for your curve. Write a sine equation for your curve.

i. What is the circumference of the wheel?

ii. At what speed is the wheel traveling? Please answer in feet / second.

iii. If you begin your ride at the base of the wheel, what is the height after 1 minute? 4 minutes?

iv. At what approximate time(s) will you reach the following heights?

a) 100 ft

b) 240 ft

v. What is the length of the arc traveled by the Navy Pier Ferris wheel from the 4 o’clock to the 7 o’clock position?

Solution to this Trigonometric Function word practice problem is provided in the video below!

## Roller Coaster trigonometry problem

A portion of a roller coaster is to be built in the shape of a sinusoid. You have been hired to calculate the lengths of the horizontal and vertical timber supports to be used.

a. The high and low points on the track are separated by 50 meters horizontally and 30 meters vertically. The low point is 3 meters below the ground. Letting y be the number of meters the track is above the ground and x the number of meters horizontally from the high point, write an equation expressing y in terms of x.

b. How long is the vertical timber at the high point? At x = 4 m? At x = 32 m?

c. Where does the track first go below ground?

Solution to this Trigonometric Function example practice problem is provided in the video below!

## Steamboat trigonometry example word problem

Mark Twain sat on the deck of a river steamboat. As the paddlewheel turned, a point on the paddle blade moved in such a way that its distance, d, from the water’s surface was a sinusoidal function of time. When his stopwatch read 4 seconds, the point was at its highest, 16 feet above the water’s surface. The wheel’s diameter was 18 feet, and it completed a revolution every 10 seconds.

a. Sketch a graph of the sinusoid

b. Write the equation of the sinusoid

c. How far above the surface was the point when Mark’s stopwatch read:

i. 5 seconds

ii. 17 seconds

d. What is the first positive value of time at which the point was at the water’s surface? At that time, was it going into or coming out of the water? Explain.

Solution to this Trigonometric Function example word problem is provided in the video below!

## Temperature trigonometry word problem

The max temperature in Buenos Aires is on January 15 and is 33 degrees Celsius. The minimum temperature is on July 16 (day 197) and is 9 degrees Celsius. (Assume the period is 365 days).

a) Sketch the temperature as a function of time

b) Find the equation for the temperature, T, as a function of time, t.

c) What is the temperature on Mother’s Day, May 10?

d) Give the dates during a one year period when the temperature is below 18 degrees Celsius.

Solution to this Trigonometric Function example word problem is provided in the video below!