Question

Cedar City is planning a third city park to be equidistant from two existing parks labeled P₁ and P₂. Below is a sketch of the city. Identify two potential locations for the third park, labeled as P₃a and P₃b on the map. Clearly outline the mathematical steps used to determine each location.

a) Use the midpoint formula to find the coordinates of the midpoints between P₁ and P₂. Label the midpoint as M.

b) Draw a circle with radius PM, where P is the center of an existing park (either P₁ or P₂), and M is the midpoint found in step (a). The intersection of these two circles gives the possible locations for the third park.

c) Repeat steps (a) and (b) for the other existing park (either P₁ or P₂).

d) The intersections in steps (b) and (c) are the potential locations for the third park, labeled as P₃a and P₃b.

Answer 1

To find the potential at P₁ and P₂, you can use either the exact formula for point charges or the approximate dipole potential formula.

Step-by-step explanation:

To find the potential at P₁ and P₂, we can use two different methods:

a) Exact formula for point charges:

1. Calculate the distance from each point to the charges using the Pythagorean theorem.

2. Use the formula V=k(q/r), where V is the potential, k is the electrostatic constant, q is the charge, and r is the distance.

3. Plug in the values to find the potential at P₁ and P₂.

b) Approximate dipole potential formula:

1. Calculate the distance from each point to the origin using the Pythagorean theorem.

2. Use the formula V=k(q/r^2) to find the potential due to each charge.

3. Add the potentials due to each charge to find the total potential.

4. Plug in the values to find the potential at P₁ and P₂.