Does the line through (3, 7) and (-2, -5) intersect with the line through (4, 8) and (10, -2)? (1 point)

Question

• 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

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 2

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 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.