I have graphed the equations onto my calculator but I don't know how to answer the questions. Can someone explain this to me?
2. Coco, the wonder cat, is watching a spider on a wall that is 18 feet wide and 12 feet tall. The spider moves along for 10 seconds along a path describe by the parametric equations
x(t)=.3(t-2.5)^2+2 and y(t)=1.5 sin t+6
if the lower left corner is taken as the origin. Coco can reach 3 feet up and can move freely from side to side (Set your calculator to Radian mode¡Xnot degree mode¡Xfor this problem).
a. What is the closest that Coco can come to the catching the spider. When does this happen? Given answers to the nearest tenth. Explain your reasoning and how you arrived at this answer.
b. How close does the spider get to the origin? Show/explain how you arrived at this answer.
daniel
(a) since Coco can be anywhere on the line y=3, you want the closest that the spider can get to that line. Since sin(t) has a minimum value of -1, y(t) has a minimum of 4.5, so Coco can never get closer than 1.5 ft to the spider.
(b) The spider's distance is
z^2 = x^2+y^2
= (.3(t-2.5)^2+2)^2 + (1.5 sin t+6)^2
z has a minimum value of about 5.5 ft
a. To find the closest distance that Coco can come to catching the spider, we need to find the distance between the path of the spider (given by the parametric equations) and the position that Coco can reach (3 feet up from the origin).
1. First, let's set up the equation for the distance between Coco and the spider's path. We can use the distance formula:
Distance = √((x_coco - x_spider)^2 + (y_coco - y_spider)^2)
where (x_coco, y_coco) are the coordinates of Coco's position and (x_spider, y_spider) are the coordinates of the spider's position at a given time.
2. Since Coco can move freely from side to side, let's consider the x-coordinate of Coco's position as the same as the x-coordinate of the spider's position (x_coco = x_spider).
3. Now, substitute the parametric equations for the spider's position into the distance formula:
Distance = √((x_spider - x_spider)^2 + (y_coco - y_spider)^2)
= √(0^2 + (y_coco - y_spider)^2)
= √(y_coco - y_spider)^2
= |y_coco - y_spider|
4. We know that Coco can reach 3 feet up (y_coco = 3) and the spider's y-coordinate is given by y_spider = 1.5 sin(t) + 6. Substituting these values into the distance equation:
Distance = |3 - (1.5 sin(t) + 6)|
= |3 - 1.5 sin(t) - 6|
= |-3 - 1.5 sin(t)|
5. To find the closest distance, we need to minimize the absolute value of this expression. Since the range of the sine function is between -1 and 1, the minimum value of the absolute value term occurs when sin(t) = -1. Therefore, we can substitute -1 into the equation:
Distance = |-3 - 1.5(-1)|
= |-3 + 1.5|
= |-1.5|
= 1.5 feet
Therefore, Coco can come closest to the spider at a distance of 1.5 feet. To find when this happens, we need to find the value of t that corresponds to sin(t) = -1. By inspecting the unit circle, we can see that sin(t) = -1 at t = 3π/2.
So, Coco can come closest to catching the spider at a distance of 1.5 feet when t = 3π/2 (rounded to the nearest tenth).
b. To find how close the spider gets to the origin, we need to find the minimum value of the x-coordinate of the spider's path.
1. Using the given parametric equation for x(t), substitute t = 2.5 into the equation:
x(t) = 0.3(t - 2.5)^2 + 2
= 0.3(2.5 - 2.5)^2 + 2
= 2
Therefore, the spider's x-coordinate is 2. To determine how close the spider gets to the origin, we need to find the distance between the spider's position and the origin:
Distance = √(x_spider^2 + y_spider^2)
= √(2^2 + (1.5 sin(t) + 6)^2)
Since we are given a specific value of t for the closest distance in part a, we can substitute t = 3π/2 into the equation:
Distance = √(2^2 + (1.5 sin(3π/2) + 6)^2)
= √(2^2 + (1.5(-1) + 6)^2)
= √(4 + (4.5 + 6)^2)
= √(4 + 10.5^2)
= √(4 + 110.25)
= √(114.25)
≈ 10.7 feet
Therefore, the spider gets closest to the origin at a distance of approximately 10.7 feet.
To answer these questions, you will need to understand how to find the distance between two points on a graph and how to read and interpret the parametric equations.
a. What is the closest that Coco can come to catching the spider? When does this happen?
To find the closest distance between Coco and the spider, you need to calculate the distance between Coco's position (x, y) and the spider's position (x(t), y(t)).
The distance formula between two points (x1, y1) and (x2, y2) is given by:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, Coco's position is at (0, 3) since the lower left corner is taken as the origin. The spider's position (x(t), y(t)) is given by the parametric equations:
x(t) = 0.3(t - 2.5)^2 + 2
y(t) = 1.5sin(t) + 6
Let's plug in these values into the distance formula:
distance = sqrt((0.3(t - 2.5)^2 + 2 - 0)^2 + (1.5sin(t) + 6 - 3)^2)
To find the closest distance, you need to minimize this distance function over the interval of t that the spider is moving. In this case, the spider moves for 10 seconds, so consider the interval t = 0 to t = 10.
You can use technology such as a graphing calculator or a computer algebra system to graph the distance function and find the minimum value. Use the graphing calculator to plot the distance function and find the minimum point. The x-coordinate of this point will give you the time (t) when Coco can come closest to catching the spider. Round the answer to the nearest tenth.
b. How close does the spider get to the origin?
To find how close the spider gets to the origin, we need to calculate the distance between the spider's position (x(t), y(t)) and the origin (0, 0).
The distance formula in this case simplifies to:
distance = sqrt((x(t) - 0)^2 + (y(t) - 0)^2)
Use the parametric equations for x(t) and y(t) given earlier and plug them into the distance formula. Simplify the equation and then find the minimum value of this distance function over the interval t = 0 to t = 10 using a graphing calculator or a computer algebra system. Round the answer to the nearest tenth.
By following these steps, you should be able to find the answers to both parts (a) and (b) of the question.