Question

Does the line through (3, 7) and (-2, -5) intersect with the line through (4, 8) and (10, -2)? (1 point)
• Yes...these lines are parallel
• No...these lines are perpendicular
• No...these lines are parallel
• Yes... these lines are perpendicular
• You cannot tell without a graph
• Yes... these lines are not parallel

Answer 1

An amusement park that you are excited to try is $28 per visit.
You have the option to purchase a monthly membership for $90 and then pay $15 for each visit.
(4 points)
You have plans to attend 3 times this summer. Which option is best?
You think you will visit probably about 8 times this summer. Which option is best?
You have budgeted $150 for visiting the park this summer.
Which option is best?
How many visits would be approximately the break even point where both options would cost about the same?
:: Pay per visit
:: Buy a membership
:: 2
:: 5
:: 7
:: 11

Answer 2

To determine whether the two lines intersect, we need to find the slopes of each line.

1. **Find the slope of the line through (3, 7) and (-2, -5)**:

\[
m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5}
\]

2. **Find the slope of the line through (4, 8) and (10, -2)**:

\[
m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = \frac{-5}{3}
\]

Now we compare the slopes:

- The slope of the first line: \( \frac{12}{5} \)
- The slope of the second line: \( \frac{-5}{3} \)

Since the slopes are not equal, the lines are not parallel.

Next, we can check if the lines are perpendicular by seeing if the product of their slopes equals -1:

\[
m_1 \cdot m_2 = \frac{12}{5} \cdot \frac{-5}{3} = \frac{12 \cdot -5}{5 \cdot 3} = \frac{-60}{15} = -4
\]

Since the product of the slopes is not -1, the lines are not perpendicular.

Since the lines are neither parallel nor perpendicular, they do intersect at some point.

Therefore, the answer is:

**Yes... these lines are not parallel**.

Answer 3

To determine the best option for visiting the amusement park, we will calculate the total costs associated with each option for different visit scenarios and find the break-even point.

**Option 1: Pay per visit**
- Cost per visit: $28
- Total cost for \( n \) visits: \( 28n \)

**Option 2: Buy a monthly membership**
- Cost of membership: $90
- Cost per visit after membership: $15
- Total cost for \( n \) visits: \( 90 + 15n \)

Now, let's analyze each scenario:

### 1. Plans to attend 3 times this summer:
- Cost using pay-per-visit: \( 28 \times 3 = 84 \)
- Cost using membership: \( 90 + 15 \times 3 = 90 + 45 = 135 \)

**Best option:** Paying $84 for per visit is cheaper.

### 2. Plans to visit about 8 times this summer:
- Cost using pay-per-visit: \( 28 \times 8 = 224 \)
- Cost using membership: \( 90 + 15 \times 8 = 90 + 120 = 210 \)

**Best option:** The membership at $210 is cheaper.

### 3. Budgeted $150 for visiting the park this summer:
- If you visit \( n \) times with the pay-per-visit option:
\[
28n \leq 150 \Rightarrow n \leq \frac{150}{28} \approx 5.36
\]
So maximum visits are 5.

- If you visit with the membership:
\[
90 + 15n \leq 150 \Rightarrow 15n \leq 60 \Rightarrow n \leq 4
\]

**Best option:** For up to 5 visits, the pay-per-visit option is better since you can visit 5 times for $140, while the membership option would limit you to 4 visits for $150.

### 4. Break-even point where both options cost the same:
We need to set the costs equal to each other:
\[
28n = 90 + 15n
\]
Subtract \( 15n \) from both sides:
\[
28n - 15n = 90
\]
\[
13n = 90 \quad \Rightarrow \quad n = \frac{90}{13} \approx 6.92
\]

Since \( n \) is approximately 6.92, rounding it suggests that the break-even point is around 7 visits.

Thus, we conclude:
- The number of visits at which both options would cost the same is **7**.