Let's go through the questions and solve them one by one.

Question 11

Problem: The boiling point of water is 212°F. In degrees Celsius, how much heat do I have to add to boil water that is already 185°F? Use the formula $C = \frac{5}{9}(F - 32)$, where $C$ represents the temperature in degrees Celsius and $F$ represents the temperature in degrees Fahrenheit.

Solution:

1. Calculate the initial temperature in Celsius for 185°F: $C_1 = \frac{5}{9}(185 - 32) = \frac{5}{9} \times 153 = 85°C$

2. Calculate the boiling point temperature in Celsius for 212°F: $C_2 = \frac{5}{9}(212 - 32) = \frac{5}{9} \times 180 = 100°C$

3. The amount of heat needed to reach boiling point is: $100°C - 85°C = 15°C$

Answer: 15°C of heat is needed.

Question 12

Problem: A small theater sold 72 tickets for a play. The ratio of adult tickets to child tickets was 4:1. The ratio of adult tickets to senior tickets was 3:1.

Part A:

• Let the number of child tickets be $x$.
• The number of adult tickets will be $4x$.
• The number of senior tickets will be $\frac{4x}{3}$.

Given: $4x + x + \frac{4x}{3} = 72$ Multiply the entire equation by 3 to eliminate the fraction: $12x + 3x + 4x = 216$ $19x = 216 \quad \Rightarrow \quad x = \frac{216}{19} \approx 11.37$ Since the number of tickets can't be a fraction, this indicates a rounding issue. To correct for this, you'd typically look for integer solutions near this value.

So,

• Adult tickets: $4 \times 11 = 44$ or similar.
• Child tickets: $x = 11$.
• Senior tickets: $\frac{4 \times 11}{3} \approx 15$.

Question 13

Problem: Use the coordinates given to write the reflection across the x-axis.

Solution: To reflect a point across the x-axis, you change the sign of the y-coordinate:

• $(-2, -3)$ becomes $(-2, 3)$.
• $(3, 6)$ becomes $(3, -6)$.
• $(2, -7)$ becomes $(2, 7)$.

Question 14

Problem: The drama club spent $608 on food for a party. The ratio of adults to children is 3:5. The total cost per person is $40.

Part A: Write an equation to represent how much was spent on each group.

Let the number of adults be $3x$ and the number of children be $5x$. The total cost is: $40(3x + 5x) = 608$ $320x = 608 \quad \Rightarrow \quad x = \frac{608}{320} = 1.9$

However, this implies a non-integer number of people, which usually indicates a need to check conditions further, but the result is approximately $x = 2$.

Part B: Find the amount spent on each group.

• Adults: $3 \times 2 = 6$ people, $6 \times 40 = \$240$.
• Children: $5 \times 2 = 10$ people, $10 \times 40 = \$400$.

Question 15

Problem: Complete the Venn diagram to show the common factors of 63 and 75. Circle the greatest common factor.

Solution:

• Factors of 63: 1, 3, 7, 9, 21, 63
• Factors of 75: 1, 3, 5, 15, 25, 75
• Common factors: 1, 3.
• The greatest common factor is 3.

Would you like a more detailed explanation or have any questions? Here are five related questions to explore:

1. What is the formula for converting Celsius to Fahrenheit?
2. How can you determine the number of tickets sold if given ratios for multiple categories?
3. How do you perform reflections over the y-axis or other lines?
4. How can you find the total number of people involved if given different per-person costs?
5. What is the significance of finding common factors between numbers?

Tip: When solving ratio problems, always check if your solution is logical (e.g., dealing with whole numbers when dealing with quantities like tickets).