consider the motion of a point (or particle) on the circumference of a rolling circle. As the circle rolls, it generates the cycloid                                              r(t) = b(ωt − sin ωt)i + b(1 − cos ωt)j where ω is the constant angular speed of the circle and b is the radius of the circle. Find the maximum speed of a point on the circumference of an automobile tire of radius 1 foot when the automobile is traveling at 60 miles per hour. Compare this speed with the speed of the automobile.

Q: Consider the following trajectory of a moving object. Find the tangential and normal components of…

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Q: Find the length of one turn of the helix given by…

A: Given: The parametric equation of the curve: .

Q: Find the speed at the given value of t. r(t) = (sin(3t), cos(6t), cos(7t)), t = %3D (플) = 프2

A: Given:

Q: Suppose an object moves in an elliptical orbit parametrized byx = 4 cos t and y = 7 sin t, where t ≥…

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Q: Find the equation of the plane perpendicular to the curve r(t) = (sin(2t), t, cos(2t)) at the point…

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Q: Consider the following parametric equation. x = 5(cos 0 + 0 sin 0) y = 5(sin 0 – 0 cos 0) What is…

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Q: Find an equation in rectangular coordinates for the curve r^2 = cos 2θ.

A: We know that, the relationship between polar and rectangular co-ordinates are given as r2=x2+y2…

Q: Find the coordinates at t = 0, π/4 , π of a particle moving along the path c(t) = (cos 2t,sin2 t).

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Q: 4. Find the length of the curve having a parametric equation of x = a cos 3 0, y = a sin 20 from 8 =…

A: We are asked to find the length of the curve whose parametric equations are as follows:…

Q: Suppose the position function of a particle is given r(t) = (√3 cos (t), t-4, √3 sin(t)). If the…

A: Given : r(t)=3cos(t), t-4, 3sin(t) is the position function of a particle. To find : If the particle…

Q: Given the acceleration, initial velocity, and initial position of a body moving along a coordinate…

A: We have given Acceleration a = 18 cos 4t Initial velocity v(0) = -7 Initial position s(0) = -13

Q: find the speed at the given value of t. r(t) = (sin 3t, cos 4t, cos 5t), t =

A: Given: rt=sin3t, cos4t, cos5t To find the speed at the given value of t=π2. Solution: Speed is the…

Q: Suppose r(t) = cos(nt) i + sin(t) j+ 3tk represents the position of a particle on a helix, where z…

A: Given the vector

Q: A slime mold grows in a spiral pattern with polar radius r=θ3r=θ3. If the mold grows by continuing…

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Q: If an object is moving along a horizontal line, such that it's velocity, v(t) = cos(3t), what is the…

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Q: Find the parametric equations of the tangent line LT to the curve C:r (t) = 2 cos(t)i + sin(t)j +…

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Q: Given parametric equations x = 2 cos 0 and y = sin 0. d²y Then dx2 in the point (V2,) is

A: Given parametric equation is  x=2cosθ and y=sinθ

Q: 3. Find T (t) and a set of parametric equations for the tangent line to the helix given by ř(t) = 2…

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Q: Describe the motion of a particle with position (x, y) as t varies in the given interval. x = 4 + 3…

A: The solution is given below :

Q: A curve is defined by the parametric equations: x(1) = e' cos t, y(t) = e' sint d²y dy and dr of the…

A: It is given that, x(t)=et cost,y(t)=et sint   It is known that, dydx can be found by chain rule and…

Q: A curve is defined by the parametric equations (t) = 8 cos(6t) and %3D y(t) 4 sin(-4t). Find . %3D

A: We have to find derivative

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A: We have to find the equation of tangent from the parametric curve

Q: Given the acceleration and initial velocity of a body moving along a coordinate line at time t, find…

A: The acceleration of a body moving along a coordinate line at time t is given by: a(t) = 8 sin(4t)…

Q: Consider the following parametric equation. #20(cos 0+0 sin 0) y- 20(sin 0-0 cos 0) What is the…

A: Given: x=20cosθ+θsinθy=20sinθ-θcosθ we know length of curveL on interval a,b is given by…

Q: Find the velocity and acceleration of a particle whose position function is <(1)=sin(21)+ cos(t)

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Q: Consider the following parametric equation. x = 7(cos 0 + 0 sin 0) y = 7(sin 0 – 0 cos 0) 11 What is…

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Q: Find the arclength L(C) of the helix C parametrized by C: r(t) = (2 cos t, -3t, 2 sin t) for 0 <t <…

A: The arc length of the parametric curve is given by: ∫ab(x'(t))2+(y'(t))2+(z'(t))2dt...(1); a≤t≤b…

Q: Find the parametric equations of the tangent line LT to the curve C:r(t) = 2 cos(t)i + sin(t)j+ 2tk…

A: A parametric equation of a curve is given by: C: r(t) = 2cos(t)i + sin(t)j + 2tk Now, we have to…

Q: following equation aU (1,1). TC with U (0,t)= 0, = 0, and U (x,0)= sin 2. %3D %3D %3D at

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Q: Find the equation of the tangent line(s) to the following set of parametric equations at the given…

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Q: Find parametric equations of the line tangent to the graph r(t) = (t cos 7t)i - (t+ t)j+ sin(t)k at…

A: Given that rt=t+cos 7ti-t2+tj+sintk   Tangents to the curve is given by Tt=ddtrt Tt=ddtt+cos…

Q: Describe the motion of a particle with position (x, y) as t varies in the given interval. x = 3…

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Q: A tangent at the pole to the polar curve r = V2 - 2 cos 0 is 0 = A

A: A tangent at the pole to the polar curve r=2-2 cos θ is, for the tangent at the pole we have,…

Q: The derivative Z of the parametric equations z= 7 sin 3t and y= cos 3t is.

A:

Q: Find T(t) and then find a set of parametric equations for the tangent line to the helix given by…

A: Consider the equation of helix given by r(t). Find the value for t , derivative at t and modulus of…

Q: Describe the motion of a particle with position (x, y) as t varies in the given interval. x = 3…

A:

Q: Find parametric equations for the line tangent to the helix r(t) =(sqrt(2)cos t)i + (sqrt(2) sin t)j…

A:

Q: The x and y coordinates of particle P, executing simple harmonic motion, are [x = A cos wt, y = A…

A: Given statement is "The x and y coordinates of particle P executing simple harmonic motion, are…

Q: x = 4 cos t and y = 4 sin t 0sts 4

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Q: Consider the following parametric equation. x = 3(cos 0+ 0 sin Ø) y = 3(sin 0 -0 cos 0) What is the…

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Q: What is the arc length of the polar curve r = 1– sin 0 for 0 <0 <? (a) 2(v2– 1) (b) 2- v2 (c) 2v2…

A: Note: Since you have asked multiple question, we will solve the first question for you. If you want…

Q: Describe the motion of a particle with position (x, y) as t varies in the given interval. x = 5 + 3…

A: Given x=5+3 cost -1   y=5+3sint -2  π2≤t≤3π2In compare 1 &amp; 2  with st.eqnx=h+r cost y=k+r sint…

Q: A particle moves along the polar curve r =1– 2 cos 0 so that = 2. Find the value of d at e = , and…

A: We are given that,                     Now,              When          , Then

Q: Find the possible values of a and b in the expression u(x, t) = cos (at) sin (bx) such that it…

A: To Find: All those values of a and b such that u(x,t)=cos (at) sin (bx) satisfies the wave equation.

Q: Find an equation of the tangent(s) to the curve with parametric equations E = t cos t, y =t sin t nt…

A: follow next step

Q: Find T(t) and a set of parametric equations for the tangent line to the helix given by 7(t) = 2…

A:

Q: Find the distance traveled by a particle with position (х, у); Cos t, y = (cos t)², 0 < t < 4TT X =

A: To solve this we have to use two point distance formula

Q: A cyclist is travelling in a straight path at a constant speed along a dirt road with a button stuck…

A: Given: The equations of a cycloid x=rt-sin t, y=r1-cos t.

Q: Consider the following parametric equation. a = 20(cos 0+0 sin 0) y = 20(sin 0-0 cos 0) What is the…

A: Consider the given information. x=20cosθ+θsinθy=20sinθ−θcosθ The formula of length of the curve in…