The position function of a particle is given by the equation 6t2 +9t where t is

Question

The position function of a particle is given by the equation 6t2 +9t where t is measured in seconds and s in meters. (a) Find the velocity at time t. (b) What is the velocity after 2 s? (c) When is the particle at rest? (d) when is the particle moving forward? (e) Draw a diagram to represent the motion of the particle. (0) Find the total distance traveled by the particle during the first five seconds. (g) Find the acceleration at time t. 56. Find the point on the line 6z +y 9 that is closest to the point (-3, 1) 57. The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2? A plane lying horizontally at an altitude of 1 mile and a speed of 500 mi/h passes directly over the radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 miles away from the station. 58. 59. Consider the function f(x)3 1. Find the coordinates of the point at which the tangent line to the curve is parallel to the line which passes through the points (1,0) and (3, 26). Draw a graph to represent this situation. 60. Find the intervals on which the following functions are increasing, decreasing, concave up. concave down; the local min and max; and also points of inflection(s). Use this information to draw the graph of the respected functions (a) f(x)4 (b) /(2) = 2c3 + 3x2-36x (c) f(x)=

Explanation / Answer

55.

s(t) = t3 - 6t2 + 9t

a) velocit v(t) = ds(t) / d(t) = d(t3 - 6t2 + 9t) / dt = d( t3 ) / dt - d(6t2) / dt +d(9t) / dt = 3t2 - 12t + 9

b) velocity after 2 seconds will be = v(2)

v(2) = 3.(22) - 12(2) + 9 = 12 - 24 + 9 = -3 m/s

c) the particle is at rest when v(t) = 0

or 3t2 - 12t + 9 = 0

or t2 - 4t + 3 = 0

(t-1)(t-3) = 0

t = 1 second and 3 seconds

d) The particle is moving forward when v(t) > 0

or 3t2 - 12t + 9 > 0

or (t-1)(t-3) > 0

t<1 or t>3

therefore, for times 0 to 1 second excluding 1 second and for times greater than 3 seconds, the particle is moving forward

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