An object is moving around the unit circle with parametric equations x(t)=cos(t)

Question

An object is moving around the unit circle with parametric equations x(t)=cos(t), y(t)=sin(t), so it's location at time t is P(t)=(cos(t),sin(t)) . Assume 0 < t < ?/2. At a given time t, the tangent line to the unit circle at the position P(t) will determine a right triangle in the first quadrant. (Connect the origin with the y-intercept and x-intercept of the tangent line.)

(a) The area of the right triangle is a(t)= 1/(2sin(x)cos(x)

Correct: Your answer is correct.

(b)

lim

t ? pi/2?

a(t)= +infinity

Correct: Your answer is correct.

(c)

lim

t ? 0+

a(t)= -infinity

Correct: Your answer is correct.

(d)

lim

t ? pi/4

a(t)= 1

Correct: Your answer is correct.

(e) With our restriction on t, the smallest t so that a(t)=2 is ??

(f) With our restriction on t, the largest t so that a(t)=2 is ??

(g) The average rate of change of the area of the triangle on the time interval [?/6,?/4] is ??

(h) The average rate of change of the area of the triangle on the time interval [?/4,?/3] is ??

(i) Create a table of values to study the average rate of change of the area of the triangle on the time intervals [?/6,b], as b approaches ?/6 from the right. The limiting value is -4/3

Correct: Your answer is correct.

(j) Create a table of values to study the average rate of change of the area of the triangle on the time intervals [a,?/3], as a approaches ?/3 from the left. The limiting value is 4/3

Correct: Your answer is correct.

Vectors Avec and Bvec lie in an xy plane. Avec has magnitude 8.80 and angle 122 degree from the positive direction of the x axis. Bvec has components Bx = -7.40 and By = -9.20. (Type 'undefined' for any answer if appropriate.) What is 7Avec · Bvec? What is 4Avec ? 3Bvec in unit-vector notation? What is 4Avec ? 3Bvec in magnitude-angle notation with spherical coordinates? What is the angle between the directions of Avec and 4Avec ? 3Bvec? (Hint: Think a bit before you resort to a calculation.) What is Avec + 4.00 k^ in unit-vector notation? What is Avec + 4.00 k hat in magnitude-angle notation with spherical coordinates?

Explanation / Answer

e)1/2sintcost=2

sintcost=1/4

sin2t=1/2

2t=pi/6 so t = pi/12

f)for min sin2t=1/2

2t=5pi/6

t=5pi/12

g)final area=1/sin2t=1/sin(pi/3)=2/1.732

initial area=sin(2*pi/4)=1

avergage change=(2/1.732-1)(pi/4-pi/6)=-0.0405

h)final area=1/sin(2*pi/3)=2/1.732

initila area=1/sin(2*pi/4)=1

change=(2/1.732-1)/(pi/3-pi/4)=0.5910

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