Lesson 8
Rising and Falling
Problem 1
A fan blade spins counterclockwise once per second.
Which of these graphs best depicts the height, \(h\), of \(P\) after \(s\) seconds? The fan blades are 1 foot long and the height is measured in feet from the center of the fan blades.
![A drawing of a fan with 5 blades. The point P lies at the end of the blade directly to the right of the center of the fan.](https://cms-im.s3.amazonaws.com/A3vGd1BaXpdWf224HC1gnuR5?response-content-disposition=inline%3B%20filename%3D%22F7-PP-fan.png%22%3B%20filename%2A%3DUTF-8%27%27F7-PP-fan.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T171757Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f9d3e8ddf87f3f867e41f025d1aec1ea905a14118cfdb53897be1f299ceca3c0)
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.
Problem 2
Which situations are modeled accurately by a periodic function? Select all that apply.
the distance from the earth to the sun as a function of time
the vertical height of a point on a rotating wheel as a function of time
the area of a sheet of paper as a function of the number of times it is folded in half
the number of centimeters in \(x\) inches
the height of a swinging pendulum as a function of time
the height of a ball tossed in the air as a function of time
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.
Problem 3
Here is the graph of a function for some values of \(x\).
- Can you extend the graph to the whole plane so that the function \(f\) is periodic? Explain your reasoning.
- Can you extend the graph to the whole plane so that the function \(f\) is not periodic? Explain your reasoning.
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.
Problem 4
- Can a non-constant linear function be periodic? Explain your reasoning.
- Can a quadratic function be periodic? Explain your reasoning.
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.
Problem 5
Do \((7,1)\) and \((\text-5,5)\) lie on the same circle centered at \((0,0)\)? Explain how you know.
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.
(From Unit 6, Lesson 1.)Problem 6
The measure of angle \(\theta\) is between 0 and \(2\pi\) radians. Which statements must be true of \(\sin(\theta)\) and \(\cos(\theta)\)? Select all that apply.
\(\cos^2(\theta) + \sin^2(\theta) = 1\)
If \(\sin(\theta) = 0\), then \(\cos(\theta) = 1\).
If \(\sin(\theta) = 1\), then \(\cos(\theta) = 0\).
\(\cos(\theta) + \sin(\theta) = 1\).
The point \((\cos(\theta),\sin(\theta))\) lies on the unit circle.
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.
(From Unit 6, Lesson 5.)Problem 7
The center of a clock is the origin \((0,0)\) in a coordinate system. The hour hand is 4 units long. What are the coordinates of the end of the hour hand at:
- 3:00
- 8:00
- 11:00
Solution
Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.
(From Unit 6, Lesson 7.)