A robotic manufacturing system uses two types of arms. Arm Alpha places square tiles, each with a side length of $5 \text{ cm}$. Arm Beta places rectangular tiles, measuring $3 \text{ cm}$ by $7 \text{ cm}$. If Arm Alpha places 10 tiles and Arm Beta places 8 tiles, what is the total area covered by all the tiles?
A circular gear component has a radius of $r = 7 \text{ cm}$. Calculate its circumference and area. (Use $\pi \approx \frac{22}{7}$)
An automated guided vehicle (AGV) is initially at coordinate P $(2, 4)$ on a factory floor grid. It receives a command to translate its position by the vector $\langle +3, -2 \rangle$. What are the new coordinates of the AGV?
A robotic arm is programmed to orient a part. It first rotates $30^\circ$ clockwise from its starting position. Then, it rotates $75^\circ$ counter-clockwise. Finally, it rotates $15^\circ$ clockwise. What is the net angular displacement (total angle and direction of rotation) of the part from its initial orientation?
A programmable lawnmower is set to mow a square patch of grass with vertices at $(1,1)$, $(5,1)$, $(5,5)$, and $(1,5)$ on a coordinate grid (units in meters). If the settings are changed to double the length of each side of the square, while keeping the bottom-left corner at $(1,1)$, what are the new coordinates of the other three vertices? What is the area of the new, larger square patch?
New coordinates for Vertex 2 (originally $(5,1)$):
New coordinates for Vertex 3 (originally $(5,5)$):
New coordinates for Vertex 4 (originally $(1,5)$):
Area of the new square patch: