029i - Problem Solving with Volume and Capacity

Lesson 29 — Problem Solving: Volume and Capacity of Right Prisms

Strand: Measurement | Descriptor: AC9M8M02 | Duration: 45 minutes

Learning Intentions

• To solve multi-step applied problems involving the volume and capacity of right prisms.
• To select and apply appropriate units, converting between volume and capacity as needed within a single solution.

Success Criteria

I can:

1. Translate a real-world scenario into a volume or capacity calculation.
2. Convert between units of volume and capacity within a single multi-step problem.
3. Solve problems involving rates, cost per unit volume, or filling and draining a container.
4. Justify whether a solution is reasonable in context.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. Doubling one dimension of a rectangular prism doubles its volume.
2. Doubling all three dimensions of a rectangular prism doubles its volume.
3. A container’s capacity in litres equals its volume in divided by .
4. Two prisms with equal volume must have the same shape.

Answers: 1. Always — volume is directly proportional to any single linear dimension when the others are fixed. 2. Never — doubling all three multiplies volume by . 3. Always — since exactly. 4. Never — a tall thin prism and a short wide one can share the same volume with completely different shapes.

Activities

Activity 1 — Guided Practice: Translating Context into Expressions (9 min)

Explicit reminder, then practice.

Remind students of the routine from Lesson 19: identify each event or quantity in order, decide what must be calculated first, then write a single plan before calculating.

I do: “A tank is filled at a rate of L per minute for minutes, then drained at L per minute for minutes. Find the volume of water remaining.”

You do: Write a single expression for each, then evaluate:

1. “A rectangular tank is filled to of its capacity. Find the volume of water in litres.”
2. “Concrete costs 180\text{m}^32.4\ \text{m}^3$. Find the total cost.”
3. “A pool holds L. A pump fills it at L per minute. Find the time, in minutes, to fill the pool from empty.”

(Answers: 1. ; 2. 4329000\div30=300$ minutes.)

Activity 2 — Applied Problem-solving (14 min)

Pairs. Every answer must include correct units and a one-sentence reasonableness check.

Problem 1. A rectangular water trough for livestock is m long, m wide, and m deep. Find its capacity in litres.

Problem 2. A concrete path has a cross-section of and is m long. Concrete costs 165\text{m}^3$. Find the total cost, to the nearest dollar.

Problem 3. A juice supplier fills mL bottles from a vat holding L. How many full bottles can be filled, and how much juice (in mL) is left over?

Problem 4. A rectangular fish tank measuring is being filled by a hose at L per minute. At the same time, a small leak drains water from the tank at mL per minute.

(a) Find the tank’s capacity in litres.

(b) Find the net rate at which the tank is filling, in L/min.

(c) Find the time to fill the tank completely, to the nearest minute.

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand: what is being asked across the three parts?	A total capacity, then a combined rate, then a time — each part depends on the one before it.
What must happen before the rates can be compared?	Both rates must be in the same unit — the leak’s rate is in mL/min, but the fill rate is in L/min.
Devise a plan for (a)	Multiply the three dimensions in cm, then convert to litres.
Carry out (a)	.
Devise a plan for (b)	Convert the leak rate to litres, then subtract it from the fill rate, since the two effects act in opposite directions.
Carry out (b)	; net rate L/min.
Devise a plan for (c)	Divide the total capacity by the net fill rate.
Carry out (c)	minutes, so 43 minutes to the nearest minute.
Looking back — why round up rather than to the nearest whole minute?	At minutes the tank is not quite full; a full extra minute is needed to reach L, so rounding up is the only sensible choice in this context.

Answers: 1 — L; reasonable, a modest trough for a few animals. 2 — ; cost 178150\ \text{L}=150,000150,000\div200=7501603.7543$ minutes.

Checks for Understanding

(8 minutes — exit ticket, collected)

1. A rectangular tank is . Find its capacity in litres.
2. A tank is filled at L/min for minutes, then drained at L/min for minutes. Find the volume of water remaining.
3. A vat holds L. Bottles of mL are filled from it. How many full bottles can be filled, and how much is left over?
4. Reasoning. A pool is filled by two hoses at once: one at L/min, the other at L/min. Explain why adding the two rates together is the correct approach, and find the combined rate.

Answers: 1. L; 2. L; 3. mL; bottles exactly, no juice left over; 4. Both hoses are filling the same pool simultaneously, so their volumes contributed per minute combine additively; combined rate L/min.

Common Misconceptions

Misconception	How to pre-empt it
Comparing or combining two rates given in different units (e.g. mL/min and L/min) without converting first.	Require every rate to be converted to a common unit before any addition or subtraction, as in Problem 4.
Rounding a “time to fill” answer down instead of up, leaving the container not quite full.	Frame these as “minimum time required” questions — check whether the rounded-down value actually reaches the target.
Forgetting that a drain or leak acts as a subtraction from a fill rate, not a separate independent quantity.	Model combined-rate problems with a single signed expression, as in Activity 1’s “I do.”
Treating “how many full containers” as a straightforward division without checking for a sensible remainder.	Always state the remainder in its own correct unit, not as a decimal fraction of a container.
Forgetting to convert all dimensions to the same unit before multiplying to find a volume.	Insist on a labelled “convert first” line whenever mixed units appear, as practised across Lessons 26–28.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A tank is filled by a pipe in minutes if used alone, or drained by a second pipe in minutes if used alone. If both pipes are open at once, starting from empty, how long does it take to fill the tank?

Answer

Let the tank’s capacity be units (chosen as the LCM of and ). Fill rate units/min; drain rate units/min. Net rate units/min. Time .

E2 (Kangaroo style). A rectangular tank is full. How many litres of water must be added to fill it completely?

Answer

Capacity L. Currently holds L. Needed: .

E3 (Challenge). Two identical rectangular tanks are each being filled at L/min. Tank A starts empty; Tank B starts with L already in it. Both tanks have capacity L. How many more minutes does Tank A take to fill than Tank B?

Answer

Tank B needs L, taking min. Tank A needs L, taking min. Difference: .

E4 (Investigation). A cylindrical rain gauge (treat as a prism with a circular cross-section of radius cm) collects mL of rain. Using and , find the depth of rainfall collected, to decimal place.

Answer

(A neat extension showing the same logic applies to circular cross-sections, previewing cylinders.)

Homework

1. A rectangular tank is . Find its capacity in litres.
2. A tank is filled at L/min for minutes, then drained at L/min for minutes. Find the volume of water remaining.
3. A drum holds L of paint. Cans of L are filled from it. How many full cans can be filled, and how much paint is left over?
4. A pool with capacity L is filled by a hose at L/min while a leak drains it at L/min. Find the net fill rate, and the time to fill the pool, to the nearest minute.
5. Reasoning. Explain why the time to fill a leaking container must always be found using the net rate, not the fill rate alone, and describe what would happen if the leak rate were greater than the fill rate.
6. Challenge. A tank is filled by Pipe A alone in minutes, or by Pipe B alone in minutes. If both pipes are used together, how long does it take to fill the tank? (Hint: choose a convenient tank capacity, such as the LCM of and .)

Answers: Q1 — L. Q2 — L. Q3 — cans exactly, no paint left over. Q4 — net rate L/min; time minutes. Q5 — the net rate accounts for water leaving the container at the same time as water entering it, so only the net rate reflects the true speed at which the container is filling; if the leak rate exceeded the fill rate, the net rate would be negative, meaning the tank would never fill and would instead empty over time. Q6 — capacity units; Pipe A rate units/min, Pipe B rate units/min; combined units/min; time minutes.