This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — here are the solutions to the next problems: 11. A rectangular prism has a base area of 24 cm^2 and a height of 5 cm. Calculate the volume of the prism using the formula: Volume = Base Area × Height. Step 1: Use the given formula for the volume of a prism. Volume = Base Area × Height Step 2: Substitute the given values into the formula. Volume = 24 cm^2 × 5 cm Step 3: Perform the multiplication. Volume = 120 cm^3 The volume of the prism is 120 cm^3 12. A swimming pool in the shape of a rectangular prism has a volume of 600 cubic metres. The length is 20 metres and the width is 15 metres. a) Determine the depth of the pool. Step 1: The volume of a rectangular prism is given by the formula: Volume = Length × Width × Depth. We are given the volume, length, and width, and need to find the depth. 600 m^3 = 20 m × 15 m × Depth Step 2: Multiply the length and width. 600 m^3 = 300 m^2 × Depth Step 3: Divide the volume by the product of length and width to find the depth. Depth = 600 m^3300 m^2 Depth = 2 m The depth of the pool is 2 m b) Explain how you used the volume formula to find the depth. I used the formula for the volume of a rectangular prism, Volume = Length × Width × Depth. I substituted the known values for the volume, length, and width, and then rearranged the formula to solve for the unknown depth by dividing the volume by the product of the length and width. c) If the pool needs to be drained and refilled, and it takes 2 hours to fill 100 cubic metres, how long will it take to fill the entire pool? Show your working. Step 1: Determine the total volume of the pool, which is 600 m^3. We are given that it takes 2 hours to fill 100 m^3. Step 2: Calculate how many 100 m^3 segments are in the total volume. Number of segments = Total VolumeVolume per segment Number of segments = 600 m^3100 m^3 Number of segments = 6 Step 3: Multiply the number of segments by the time it takes to fill one segment. Total time = 6 × 2 hours Total time = 12 hours It will take 12 hours to fill the entire pool. 13. A school collected data on student attendance over a term. The data shows (in percentages): 95, 92, 88, 95, 90, 87, 95, 91. a) Calculate the mean, median, and mode for this data. Step 1: Order the data from least to greatest: 87, 88, 90, 91, 92, 95, 95, 95. There are n=8 data points. Step 2: Calculate the mean. Mean = (87+88+90+91+92+95+95+95)/(8) Mean = (733)/(8) Mean = 91.625 Step 3: Calculate the median. Since there are 8 data points (an even number), the median is the average of the two middle values (the 4th and 5th values). The ordered data is: 87, 88, 90, 91, 92, 95, 95, 95. Median = (91+92)/(2) Median = (183)/(2) Median = 91.5 Step 4: Calculate the mode. The mode is the value that appears most frequently. In the data set (87, 88, 90, 91, 92, 95, 95, 95), the value 95 appears three times, which is more than any other value. Mode = 95 The mean is 91.625, the median is 91.5, and the mode is 95 b) Analyse which measure of central tendency best represents the typical attendance rate. The median or the mean would best represent the typical attendance rate. The mode (95%) is the most frequent attendance, but it might not be typical if other values are spread out. The mean (91.625%) gives an average, while the median (91.5%) represents the middle value, less affected by extreme values if they were present. Given the data, all three are quite close. c) Justify your choice with reference to the data. I would choose the median (91.5%) as the best representation. The data set is fairly symmetrical without extreme outliers, and the median is robust to any potential skewness or outliers. While the mean (91.625%) is very close, the median provides a clear central point of the ordered data. The mode (95%) is the most frequent, but it's at the higher end of the data, and not necessarily the "typical" central value. Drop the next question!