--- title: "Prism problems" subtitle: "Area, volume, circles and Pythagoras · Volume · Lesson 15" --- # Prism problems **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A farmer needs a water trough that holds 500 L, a builder orders concrete in cubic metres, a pool needs filling. Each is a prism problem, often run backwards. :::teacher-note **In one sentence:** Students use V = Ah for rectangular and triangular prisms, work backwards to missing lengths, and solve capacity problems in context (AC9M8M02, node 2). **Students should be able to say:** I can find the volume of any right prism from the area of its end. I can find a missing length from a volume or a capacity. I can answer in the units the problem asks for. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Change the litres to cubic metres first. :::keep-together **A tank has a base 4 m by 3 m and holds 36 000 L when full. How deep is it?** ☐ 0.3 m ☐ 3 m ☐ 12 m ☐ 3000 m My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 3 m. 36 000 L = 36 m³. The base is 4 × 3 = 12 m², so the depth is 36 ÷ 12 = 3 m. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## The end, then the length :::keep-together ### Any prism Here is the routine. 1. $A = \tfrac{1}{2} \times 6 \times 4 = 12\ \text{cm}^2$ — Find the area of the end with its own formula. 2. $V = A \times h = 12 \times 10 = 120\ \text{cm}^3$ — Times the length of the prism. 3. $h = V \div A$ — To find a missing length, divide the volume by the end's area. Now try one. **A triangular prism's end has legs 5 cm and 4 cm (a right angle between them). The prism is 9 cm long. What is its volume in cm³?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 90 A = ½ × 5 × 4 = 10 cm², so V = 10 × 9 = 90 cm³. ::: :::keep-together ### Find the mistake **Mia finds the volume of a triangular prism. Click the line where her working goes wrong.** ``` 1 A prism's end is a right triangle with legs 3 cm and 4 cm. The prism is 10 cm long. 2 Mia uses volume = length × width × height. 3 V = 3 × 4 × 10 = 120 cm³. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: l × w × h is for a rectangular box. Any prism is the area of its end times its length: ½ × 3 × 4 × 10 = 60 cm³. ::: :::keep-together ### What you need **The end of a triangular prism is a right triangle with sides 3 cm, 4 cm and 5 cm. Which do you need for its area?** ☐ All three sides ☐ The 3 cm and 4 cm sides (they meet at the right angle) ☐ Only the 5 cm side ☐ The 5 cm side and the length ::: :::teacher-note **Answer:** The 3 cm and 4 cm sides (they meet at the right angle) The two legs are a base and its perpendicular height: ½ × 3 × 4 = 6 cm². ::: :::typical ## Work backwards **A volume of 60.** **Find the height.** **Fill a pool.** **A composite end.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Prism problems :::keep-together ### End area times length **A prism's end has an area of 15 cm² and it is 8 cm long. What is its volume in cm³?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 120 15 × 8 = 120 cm³. ::: :::keep-together ### A fish tank **A fish tank is 50 cm by 40 cm by 30 cm. How many litres does it hold?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 60 50 × 40 × 30 = 60 000 cm³ = 60 L. ::: :::keep-together ### Buckets **A trough is 2 m long, 1 m wide and 0.5 m deep. How many 10 L buckets fill it?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 100 2 × 1 × 0.5 = 1 m³ = 1000 L, and 1000 ÷ 10 = 100 buckets. ::: :::keep-together ### Review: litres **Review: how many mL does 1.25 L hold?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1250 1.25 × 1000 = 1250 mL. ::: ## Make: a box to order Then explain how you found the height before you moved the slider. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A tank with a 2 m by 1.5 m base holds 4500 L. How deep is the water?** {{lines-3}} ::: :::teacher-note **Answer:** 4500 L = 4.5 m³; 4.5 ÷ 3 = 1.5 m. :::