I’m working on a mathematics writing question and need an explanation to help me understand better.
Transcript: Point on a Circle
The video begins with a woman with glasses standing to one side of the screen.
Hi, I’m Joy. I’m helping my cousin David with his new business. It’s a pony ride business. [A picture is shown of a young boy on a pony behind a gate. On the gate is a sign that says “Pony Rides $3.00.”] Yes, really.
David’s taking Cupcake that’s the name of one of his ponies to a birthday party for a customer in their garden. [A picture of a white miniature pony is shown.] Luckily for David, I know my geometry. Here’s the problem.
The customer has a rectangular yard with a fence on two sides. [An image of a yard is shown with a fence along the bottom and right side of the yard.] There’s one precious rose bush that Cupcake must not trample. It’s located at a point 16 feet over, and 18 feet up. [A red dot is placed in the yard. Above the red dot is written “Rose bush at point (16, 18).]
David’s going to put Cupcake on a 10 foot rope, and Cupcake will walk in a circular path that’s the length of the rope. [A figure is shown of a circle with a radius marked as 10 feet. Beneath the center point is written “Center stake??.”] But where do we put the stake? [The circle moves around the yard.]
David says if a stake is put at a point that is 10 feet over, and 10 feet up from the bottom corner, the rose bush will be safe. [A star labeled “David’s idea (10, 10) is placed in the yard.] I think the rose bush will get trampled.
I say we need to put the stake at a point that is 8 feet over, and 8 feet up. [A star labeled “Joy’s idea (8, 8)” is placed below the first star.] David thinks if we put it here, Cupcake is going to hit the fence. Um, why would a pony walk into a fence?
[The video returns to show Joy.] So, who is correct? Where should we put the stake to make sure Cupcake doesn’t trample the rose bush or run into the fence? [A picture is shown of Cupcake wearing a bridle.] And why am I spending my Saturday this way? [The sound of a horse whinnying is heard in the background.]
Scenario: Pony Ride
- View the video found on page 1 of this journal activity.
- Using the information provided in the video, answer the questions below.
- Show your work for all calculations
The Students Conjectures: The students are running a pony ride business. They are trying to decide where to stake the pony so that the pony can’t reach the fence or trample a rose bush. The rose bush is located 16 feet over and 18 feet up from the bottom left corner. That can be represented by (16, 18). David is putting the pony on a 10-foot rope.
1. Complete the table below to summarize what you know about each students conjecture: (2 points, 1 point for each row of the chart)
Analyze the Conjecture:
2. What are the arguments against each student’s plan? (1 point)
Analyze the Data: Suppose that this grid represents the customers yard. The bottom left corner is the origin point, (0, 0), and the x- and y- axes represent the two fences. The rose bush is at point (16, 18).
Use this graph for questions 3 through 13.
3. What are the coordinates of the point where David wants to put the stake? Draw this point on the graph. (1 point)
4. If they stake the pony where David wants, what is the equation of the circle that represents the ponys path? (2 points)
a) What will be the radius of the ponys path? (1 point)
b) Remember that the equation of a circle whose center is not at the origin is
5. Sketch the circle represented by this equation onto the graph. (1 point)
6. To check whether the rose bush lies on the circle, see if its coordinates make the equation for the circle true. Show your work.(2 points)
7. What are the coordinates of the point where Joy wants to put the stake? Draw this point on your graph. (1 point)
8. If they stake the pony where Joy wants, what is the equation of the circle that represents the ponys path? (2 points)
9. Sketch the circle represented by this equation on the graph. (1 point)
10. To check whether the rose bush lies on the circle, see if its coordinates make the equation for the circle true. Show your work. (2 points)
11. If the pony is staked at the point (8,8), is the pony able to move about freely without running into the fence? Explain your answer using the location of the stake and the length of the rope to support your reasoning. (1 point)
Making a Decision
12. Who was correct? Is either point (10, 10) or (8, 8) a good place to stake the pony? (1 point)
13. Consider the circle with equation (x 12)2 + (y 12)2 = 100.
a) What is the center of this circle? (1 point)
b) Would this circle work for the pony’s path? Why or why not? Explain your answer by using the equation, the location of the stake, and the length of the rope to support your reasoning. (2 points)