Question 1: A survey recorded the heights (in cm) of 30 architects' designs for building columns. The cumulative frequency table shows the number of columns up to certain heights: | Height (cm) | Cumulative Frequency | |--------------|-------------------------| | 50 | 5 | | 60 | 12 | | 70 | 20 | | 80 | 27 | | 90 | 30 | Calculate the number of columns measuring between 60 cm and 80 cm.

Answer: 15

Question 2: Construct a cumulative frequency table for the following data on the lengths of beams used in bridge construction (in meters): 10, 12, 11, 13, 12, 14, 15, 12, 11, 13.

Answer: See working for cumulative table creation

Question 3: A design team is analyzing the span lengths of beams used in various building projects. The cumulative frequency table indicates that 8 beams are less than or equal to 10 meters, and 15 are less than or equal to 12 meters. How many beams are between 10 and 12 meters?

Answer: 7

Question 4: Explain how cumulative frequency helps architects decide on the most common beam length used in their projects.

Answer: It shows the total number of beams up to each length, helping identify the most frequently used length.

Question 5: The table below shows the cumulative frequency of window sizes (in cm) in a new building design: | Size (cm) | Cumulative Frequency | |------------|-------------------------| | 100 | 4 | | 110 | 9 | | 120 | 15 | | 130 | 20 | | 140 | 25 | Determine the total number of window sizes recorded.

Answer: 25

Question 6: A survey records the interior wall lengths (in meters) of 40 different rooms. The cumulative frequency table is: | Length (m) | Cumulative Frequency | |--------------|-------------------------| | 3 | 8 | | 4 | 20 | | 5 | 30 | | 6 | 36 | | 7 | 40 | Calculate how many rooms have interior wall lengths between 4 and 6 meters.

Answer: 16

Question 7: In a project to design curved archways, the lengths (in cm) of the arches follow a cumulative frequency pattern. If 22 arches are less than or equal to 150 cm, and 35 arches are less than or equal to 180 cm, how many arches measure between 150 cm and 180 cm?

Answer: 13

Question 8: Describe one way that cumulative frequency diagrams can assist architects during the planning phase of a building project.

Answer: They help visualize the distribution of measurements, making it easier to choose most common sizes for materials.

Question 9: Using the data from the previous question, what is the approximate number of arches measuring more than 180 cm?

Answer: 5

Question 10: Identify a common mistake students might make when constructing a cumulative frequency table and explain how to avoid it.

Answer: Adding individual data instead of cumulative totals; ensure each total includes all previous data.

Question 11: Challenge: Given a cumulative frequency table for window sizes, interpret the data to estimate the most common size range and justify your reasoning.

Answer: The most common size range is where the greatest increase occurs in the table, indicating the mode.