Final answer:
The correct equation to find the number of people (p) for which the cost of the waterpark and the rock-climbing gym are the same is 45p = 150 + 20p. After simplifying and solving the equation, we find that for 6 people, both activities cost the same. Therefore, the correct option in the final part of the question is (a) 40p + 5 = 150 + 20p
Step-by-step explanation:
To find p, the number of people for which the cost of the waterpark and the rock-climbing gym are equal, we need to set up an equation where the costs of both activities are the same. The waterpark charges $40 per person for admission and $5 per person for tube rentals, giving us a cost of 40p + 5p for p people. The rock-climbing gym charges a flat $150 for facility rental plus $20 per person, which gives us an additional cost of 20p. Thus, the equation to find the equality in costs is 40p + 5p = 150 + 20p.
Combining the like terms on the waterpark's side of the equation leads to 45p representing the total cost for p people at the waterpark. Now the equation looks like 45p = 150 + 20p. To isolate p and solve for it, you subtract 20p from both sides of the equation, which leads to 25p = 150 after simplification. Lastly, divide both sides by 25 to get p = 6, which means that for 6 people, the total cost of going to the waterpark is the same as going to the rock-climbing gym.