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 and 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—not degree mode—for 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.
Answer:
Step-by-step explanation:
a. What is the closest that Coco can come to the catching the spider. When does this happen?
From the given information;
Coco can reach 3 feet up and can move freely from side to side; thus as long as Coco can move freely about any region on the line y =3 , thus we want the closest that Coco can come to the catching the spider at that line.
So; Sin(t) has a minimum value of -1
y(t) has a minimum value of 4.5
Thus; the closest that Coco can come to the catching the spider is :
y = (4.5 -3) ft
y = 1.5 ft
b. How close does the spider get to the origin?
The spider location away from the origin is:
z² = x² + y²
x(t)=0.3(t-2.5)^2+2
y(t)=1.5 sin t+6
∵
z² = (0.3(t-2.5)²+2)² + ( 1.5 sin t + 6)²
As we know that z will be minimum when z² is minimum; then:
[tex]\dfrac{dz}{dt}=0 \Longrightarrow 2(0.3 (t - 2.5)^2+2) \times 0.6(t-2.5)+2(1.5 \ sin t+ 6) \times 1.5 cos \ t[/tex]
z = 5.5 ft
So, z is minimum at about 5.5 ft.
Thus; the spider is 5.5 ft close to the origin.
