Final answer:
The coordinates for the bear exhibit are (14, 1), using the midpoint formula based on the provided coordinates for the tiger exhibit and the snack bar. The treasure hunter's findings, including silver and gold coins, can be described using polar and rectangular coordinate systems.
Step-by-step explanation:
To determine the coordinates for the bear exhibit given that the snack bar is the halfway point between the tiger exhibit and the bear exhibit, we use the midpoint formula.
The midpoint (M) between two points, A (x1, y1) and B (x2, y2), in a Cartesian coordinate system is given by M = ( (x1 + x2)/2, (y1 + y2)/2 ).
The tiger exhibit is at (10, 15) and the snack bar, our midpoint, is at (12, 8).
We need to solve for the coordinates of the bear exhibit, let's call it point B (x, y). Since (12, 8) is the midpoint, apply the formula as follows:
- x = 2 × 12 - 10 = 14
- y = 2 × 8 - 15 = 1
Therefore, the coordinates for the bear exhibit are (14, 1).
Regarding polar coordinates, a treasure hunter's findings can be described using both polar and rectangular coordinates relative to a reference point. If the silver coin is 20.0 m away from a well in the direction 20° north of east, the polar coordinates would be (20.0 m, 20°).
To convert this to rectangular coordinates, we use the equations x = r cos(θ) and y = r sin(θ), where r is the distance from the origin and θ is the angle with respect to the positive x-axis.
The gold coin is 10.0 m away in the direction 20° north of west which would have polar coordinates (10.0 m, 160°), since we typically measure angles in a counter-clockwise direction from the positive x-axis.