11.4.1Let's get ready to rumble!

Battling Robots

11-103.

BATTLING ROBOTS, Part One

LET’S GET READY TO RUMBLE!

Congratulations! You are one of four finalists in the Battling Robot competition. You will have one minute to earn as many points as possible by hitting your opponents. The robot with the most points at the end will be crowned Robot Supreme!

Beware of the Buzz Saws! If your robot runs into one of the Buzz Saws, your robot loses points and risks being sliced to bits. However, robots that hit an Energy Boost are awarded bonus points and are re-energized.

The competition arena is $100$ feet long by $50$ feet wide. As the competition takes place, points will be awarded or deducted for the following:

	POINTS
Collision—your robot has the greater speed	$+200$
Collision—your robot has the lesser speed (Ouch!)	$–50$
Going out of bounds	$–100$
Energy Boost	$+25$
Buzz Saw	$–25$

You will be assigned a robot. The parametric equations for each robot are listed below (t is in seconds). Your task is to analyze your robot’s path and simulate this experience on both your graphing calculator and the Lesson 11.4.1B Resource Page. Ready… Set… GO! This can be done using the Robot Battle eTool (Desmos).

Robot	Equation	Robot	Equation
Speedy	$\begin{equation} \left\{ \begin{array}{@{}ll@{}} x(t)=36\text{cos}(\frac{\pi t}{18}) \\ y(t)=18\text{sin}(\frac{\pi t}{18}) \end{array}\right. \end{equation}$	Mach One	$\begin{equation} \left\{ \begin{array}{@{}ll@{}} x(t)=-9-21\text{cos}(\frac{\pi t}{24}) \\ y(t)=-10-14\text{cos}(\frac{\pi t}{24}) \end{array}\right. \end{equation}$
Robot $\mathbf{X}$	$\begin{equation} \left\{ \begin{array}{@{}ll@{}} x(t)=2t-36 \\ y(t)=-4\text{cos}(\frac{\pi t}{8}) \end{array}\right. \end{equation}$	Herby	$\begin{equation} \left\{ \begin{array}{@{}ll@{}} x(t)=\frac{3t}{4}\text{cos}(\frac{\pi t}{18}) \\ y(t)=\frac{3t}{8}\text{sin}(\frac{\pi t}{18}) \end{array}\right. \end{equation}$

11-104.

BATTLING ROBOTS, Part Two

Now that you have seen a simulation of the competition, it is time to determine which robot wins and to further study the motion of your robot.

For each collision that occurs, verify that the robots hit each other and determine the speed of both robots at the time of collision.
Determine the number of points each robot earns. Which robot wins the contest to become Robot Supreme?

11-105.

BATTLING ROBOTS, Part Three

Each robot’s speed changed throughout the contest. Discuss with your team which graphing mode (rectangular, parametric, polar, etc.) is appropriate for graphing speed as a function of time.

For each robot, graph its speed as a function of time using a graphing calculator. Sketch the resulting graphs on your paper.
Determine each robot’s maximum speed, total distance traveled, and average speed during the contest.
Record several observations that your team makes about the speed and distance traveled for the four robots.

11-106.

You are designing a new robot to win the Battling Robot contest. The four original robots are going to move along the same paths as given in problem 11-103. Design a path and provide the accompanying parametric equations so that your robot will win and be crowned Robot Supreme.

11-107.

Write the derivative, $\frac { d y } { d x }$ , of each equation below. Homework Help ✎

$x \sin(y) - 10y^2 = y \ln(x)$

$r = 2 - \cos(θ)$

$\left\{ \begin{array} { l } { x ( t ) = 4 t - t ^ { 2 } } \\ { y ( t ) = \operatorname { cos } ( 2 t ) } \end{array} \right.$

$y = x \sec\sqrt { 2 x ^ { 3 } - 4 x }$

11-108.

Compute without a calculator No calculator! The velocity of a train during a $90$ minute ride is given by $v = e^{2t} + 34$ miles per hour where $t$ is in hours. What is the average velocity during the trip? Homework Help ✎

$\$

11-109.

Multiple Choice: Which differential equation below is a solution to $\frac { d y } { d x }= y^2(2t + 3)$ ? Homework Help ✎

$y = −\frac { 1 } { t ^ { 2 } + 3 t + 2 }$

$y =\frac { 1 } { t ^ { 2 } + 3 t + 2 }$

$y = e^{−(t^2+3t+2)}$

$y = e^{t^2+3t+2}$

None of these.

11-110.

Multiple Choice: Based on the graph of $y = f ′\left(x\right)$ shown at right, $f$ must have a relative minimum at: Homework Help ✎

$x = 0$

$x = 2$

$x = 5$

$x = 6$

$x = 2$ and $6$

Continuous curve labeled, f prime of x, coming from left just below x axis, turning down @, the origin, turning up @, (1.5, comma negative 0.25), passing through (2, comma 0), turning down @, (3, comma 0.5), passing through (4 comma 0), turning up @, (5, comma negative 1), passing through (6, comma 0), continuing up & right.

11-111.

A projectile travels through the air with position $x\left(t\right) = 15t$ and $y\left(t\right)=-4.9t^2+46t+11$ where $x$ is horizontal displacement and $y$ is height above the ground where both are measured in meters and $t$ is in seconds. Homework Help ✎

When does the projectile hit the ground?
How far does the projectile travel during its trip through the air? That is, calculate the length of the path of the projectile.
What is the maximum height of the projectile?
What are the acceleration and velocity vectors of the projectile at time $t = 2$ seconds?

11-112.

Multiple Choice: If $F(x)=\int_a^xf(t)dt$ , then $\frac{d}{dx}F(3x^2)=\$ Homework Help ✎

$6xf(x)$

$6xf(3x^2)$

$3x^2F(6x)$

$6x F(3x^2)$

$6x f(3t^2)$

11-113.

Multiple Choice: $\int \frac { 3 } { ( 2 x - 1 ) ( x + 5 ) } d x=$ Homework Help ✎

$\frac { 3 } { 11 } \operatorname { ln } | \frac { 2 x - 1 } { x + 5 } |+ C$

$\frac { 6 } { 11 } \operatorname { ln } | 2 x - 1 | - \frac { 3 } { 11 } \operatorname { ln } | x + 5 |+ C$

$\frac { 1 } { 2 }\ln| ( 2 x - 1 ) ( x + 5 ) |+C$

$\operatorname { ln } | 2 x - 1 | + \operatorname { ln } | x + 5 |+C$

None of these

11-114.

Multiple Choice: A point traveling along the $x$ -axis has position $s(t) = 2t^3 - 4t^2 + 2t + 1$ . The total distance traveled by the point from $t = 0$ to $t = 2$ is: Homework Help ✎

$4.593$

$6.296$

$6.593$

11-115.

Multiple Choice: The luminous intensity $E$ of a light bulb, measured in lumens/ft², varies inversely as the square of the distance $s$ from the bulb. $E = 5.2$ lumens/ft² when $s = 5 \text{ ft}$ for a $100$ watt bulb. If you are moving away from a $100$ watt bulb at a speed of $2$ ft/sec and you are $3$ feet from the bulb, the luminous intensity is changing at the rate of: Homework Help ✎

$- \frac { 520 } { 27 }$ lumens/ft²

$- \frac { 260 } { 27 }$ lumens/ft²

$- \frac { 130 } { 27 }$ lumens/ft²

$- \frac { 130 } { 9 }$ lumens/ft²

$- \frac { 260 } { 9 }$ lumens/ft²

11-116.

Multiple Choice: The graph of $y = f(x)$ is shown at right. The function is twice differentiable. Examine the graph and decide which statement below is true. Homework Help ✎

$f(2) < f ′(2) < f ″(2)$
$f(2) < f ″(2) < f ′(2)$
$f ′(2) < f(2) < f ″(2)$
$f ″(2) < f(2) < f ′(2)$
$f ″(2) < f ′(2) < f(2)$

Calculus Third Edition