Backyard Pool Fill: Algebra and Measurement
A homeowner wants to fill a rectangular pool that is 8 m long, 4 m wide, and has a constant depth of 1.5 m. The garden hose delivers water at a rate of 12 litres per minute. However, the pool has a small leak that loses water at a rate of 0.8 litres per minute. The homeowner starts filling the pool at 9:00 AM. Water costs $2.50 per kilolitre (1 kL = 1000 L). The homeowner wants to know when the pool will be full and how much the water will cost, assuming the leak continues at a constant rate. Additionally, the homeowner plans to add a chemical that requires a dose of 5 mL per 1000 L of pool volume. The chemical is sold in 200 mL bottles.
1. Calculate the volume of the pool in litres and determine the net filling rate. Then find the time (in hours and minutes) when the pool will be full.
Read the worked answer
Volume = 8 m × 4 m × 1.5 m = 48 m³. Since 1 m³ = 1000 L, volume = 48,000 L. Net filling rate = 12 L/min - 0.8 L/min = 11.2 L/min. Time to fill = 48,000 L ÷ 11.2 L/min = 4285.714... minutes. Convert to hours: 4285.714 ÷ 60 = 71.4286 hours, which is 71 hours and 0.4286×60 ≈ 25.7 minutes, so about 71 hours 26 minutes. That is nearly 3 days later. Starting at 9:00 AM, the pool will be full at approximately 9:00 AM + 71 h 26 min = 9:00 AM on the third day plus 23 h 26 min? Let's compute: 71 h = 2 days (48 h) + 23 h. So 9:00 AM + 2 days = 9:00 AM two days later; then +23 h = 8:00 AM the next day; then +26 min = 8:26 AM. So full at about 8:26 AM on the fourth day. (Or simply state 71 h 26 min after 9:00 AM.)
2. Calculate the cost of the water used to fill the pool, and determine how many bottles of chemical are needed and the total cost if each bottle costs $12.50.
Read the worked answer
Water volume = 48,000 L = 48 kL. Cost = 48 kL × $2.50/kL = $120.00. Chemical dose: 5 mL per 1000 L. For 48,000 L, dose = 5 × (48,000/1000) = 5 × 48 = 240 mL. Bottles needed: 240 mL ÷ 200 mL/bottle = 1.2, so must buy 2 bottles. Total chemical cost = 2 × $12.50 = $25.00. Total cost = $120.00 + $25.00 = $145.00.