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Molly and her friends are planning a weekend vacation to an island that is about 90 minutes away. They have purchased non-refundable tickets to a concert and a local food tasting that only happens once each year.

After reading about the expected conditions on the water, Molly realizes she and her friends will need to sail through some rough waters. She has mapped out the two best routes:

- The height of the waves, in feet, on Route 1 can be modeled by [tex] f(x) = -9 \cos \left(\frac{\pi}{15}(x-3) + 8\right) [/tex].
- The height, in feet, of the waves on Route 2 can be modeled by [tex] g(x) = -15 \cos \left(\frac{\pi}{24}(x-15) + 10\right) [/tex].

For both functions, the value of [tex] x [/tex] measures the number of minutes since the boat has left shore. Molly expects the water traffic to be heavier along Route 2.

Molly is nervous about sailing through these waters in her current watercraft, which is a 24-foot cabin cruiser. This craft has a completely enclosed pilot house that allows Molly to stay out of the elements as she drives the boat. It also has a completely enclosed cabin that will let her friends stay warm and dry on the trip. However, it is not the most maneuverable in rough waters. Molly knows that her boat can easily handle waves that have a height less than or equal to about [tex] \frac{1}{3} [/tex] of the boat's length.

One of Molly's friends suggests that she borrow her uncle's boat. It is 30 feet long but does not have an enclosed area for the guests or the pilot; it has only a canopy that covers the piloting area.



Answer :

GinnyAnswer
To determine which boat is more suitable for Molly and her friends' trip to the island, we will assess both Route 1 and Route 2 for wave heights and see if they exceed the maximum limit for Molly's boat (24-foot cabin cruiser) and the uncle's boat (30-foot boat).

### Boat Wave Limits:
First, let's calculate the maximum allowable wave heights for both boats:

1. Molly's current boat:
[tex]\[ \text{Boat 1 Length} = 24 \text{ feet} \][/tex]
The maximum wave height it can handle:
[tex]\[ \text{Wave Limit for Boat 1} = \frac{24}{3} = 8 \text{ feet} \][/tex]

2. The uncle's boat:
[tex]\[ \text{Boat 2 Length} = 30 \text{ feet} \][/tex]
The maximum wave height it can handle:
[tex]\[ \text{Wave Limit for Boat 2} = \frac{30}{3} = 10 \text{ feet} \][/tex]

### Route 1 Wave Height:
The wave height on Route 1 is given by the function:
[tex]\[ f(x) = -9 \cos \left(\frac{\pi}{15}(x-3) + 8\right) \][/tex]
We need to check if the wave height on this route ever exceeds the limits of either boat during the 90-minute trip.

For Molly's boat (Boat 1):
[tex]\[ f(x) \text{ exceeds } 8 \text{ feet} \][/tex]

For the uncle's boat (Boat 2):
[tex]\[ f(x) \text{ does not exceed } 10 \text{ feet} \][/tex]

So, on Route 1:
- Molly's boat is not suitable as it exceeds the wave limit.
- The uncle's boat is suitable.

### Route 2 Wave Height:
The wave height on Route 2 is given by the function:
[tex]\[ g(x) = -15 \cos \left(\frac{\pi}{24}(x-15) + 10\right) \][/tex]
We need to check if the wave height on this route ever exceeds the limits of either boat during the 90-minute trip.

For Molly's boat (Boat 1):
[tex]\[ g(x) \text{ exceeds } 8 \text{ feet} \][/tex]

For the uncle's boat (Boat 2):
[tex]\[ g(x) \text{ exceeds } 10 \text{ feet} \][/tex]

So, on Route 2:
- Molly's boat is not suitable as it exceeds the wave limit.
- The uncle's boat is not suitable as well for this route since it also exceeds the wave limit.

### Conclusion:
Based on the wave height calculations:
- Route 1: Molly's boat is not suitable, but the uncle's boat is suitable.
- Route 2: Both boats are not suitable due to exceeding wave height limits.

Therefore, if Molly and her friends want to proceed with this trip, they should take Route 1 and use the uncle's 30-foot boat.

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