AP® Calculus BC · Free response

A robot on a gym floor: speed, position and distance

write it first, then score yourself point by point

Parametric, vector or polar functions · Part A: graphing calculator allowed · 9 points

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The question

Name: ______________________   Date: ____________   A robot on a gym floor: speed, position and distance (9 points)

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A robot moving on a gym floor

A robot moves on a flat gym floor, set up as the xyxy-plane with distances in meters. For 0≤t≤50 \le t \le 5 seconds, its velocity components are given below, and at t=0t = 0 the robot is at the point (1,2)(1, 2). A graphing calculator is required.
Velocity components of the robot, in meters per second, and its position at t=0t = 0
QuantityValue
dxdt\frac{dx}{dt}4sin⁡(0.7t)−14\sin(0.7t) - 1 meters per second
dydt\frac{dy}{dt}2cos⁡(0.6t)+0.52\cos(0.6t) + 0.5 meters per second
(x(0),y(0))(x(0), y(0))(1,2)(1, 2)

  1. (a) Find the speed of the robot at time t=4t = 4. Find the acceleration vector of the robot at time t=4t = 4. [2 points]

  2. (b) Find all times tt, for 0<t<50 < t < 5, at which the speed of the robot is 3 meters per second. [2 points]

  3. (c) Find the position of the robot at time t=5t = 5. [3 points]

  4. (d) Find the total distance traveled by the robot over the time interval 0≤t≤50 \le t \le 5. [2 points]