A travel agency sells ship cruises for a popular cruises line. Historically, 135 cruises can be sold when the price is $950 per person. If the price drops to the minimum allowed by the cruises line of $725 per person, 180 cruises can be sold. If the number of cruises sold increase at a constant rate as the price p decrease, where `pge725`, which of the following functions best models the situation described?

Options

A. `f(p)=-(1)/(29)p+205`
B. `f(p)=-(1)/(19)p+1,135`
C. `f(p)=-5p+4,885`
D. `f(p)=-(1)/(5)p+325`

Answer

To solve the problem, we need to find a linear function that models the relationship between the price of the cruise (p) and the number of cruises sold (N). We have two points based on the information provided:

1. When the price is $950, the number of cruises sold is 135.
2. When the price is $725, the number of cruises sold is 180.

Let's denote the price as \( p \) and the number of cruises sold as \( N \). We can represent the two points as:
- Point 1: \( (950, 135) \)
- Point 2: \( (725, 180) \)

### Step 1: Calculate the slope (m) of the line
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we will use:
- \( (x_1, y_1) = (950, 135) \)
- \( (x_2, y_2) = (725, 180) \)

Substituting the values:
\[ m = \frac{180 - 135}{725 - 950} = \frac{45}{-225} = -\frac{1}{5} \]

### Step 2: Use point-slope form to find the equation
The point-slope form of a linear equation is:
\[ y - y_1 = m(x - x_1) \]
Using point \( (950, 135) \) and the slope \( m = -\frac{1}{5} \):
\[ N - 135 = -\frac{1}{5}(p - 950) \]

### Step 3: Rearranging to slope-intercept form
Now, we will rearrange the equation to solve for \( N \):
\[ N - 135 = -\frac{1}{5}p + \frac{950}{5} \]
Calculating \( \frac{950}{5} \):
\[ \frac{950}{5} = 190 \]
So the equation becomes:
\[ N - 135 = -\frac{1}{5}p + 190 \]
Adding 135 to both sides:
\[ N = -\frac{1}{5}p + 190 + 135 \]
\[ N = -\frac{1}{5}p + 325 \]

### Step 4: Final equation
Thus, the function that models the situation is:
\[ N(p) = -\frac{1}{5}p + 325 \]