Scaling a fish tank: volume, surface area and a misleading ratio
A student designs a rectangular fish tank 60 cm long, 30 cm wide and 40 cm high, with glass 0.5 cm thick (ignore thickness for volume calculations but not for a later question). A pet-shop guide says a tank should provide "at least 1 litre of water per 1 cm of fish length" and that surface area should be "about 1.5 times the tank's length in cm² per cm of fish". The student scales every dimension up by a factor of 1.5 to "give the fish more room" and claims the water volume increases by 1.5 times. The guide's surface-area rule is ambiguous, and the student must decide what it actually constrains.
1. Compute the original volume in litres and the volume after scaling all dimensions by 1.5, then correct the student's claim.
Read the worked answer
Original volume = 60 × 30 × 40 = 72,000 cm³ = 72 L.
Scaled dimensions: 90 × 45 × 60 = 243,000 cm³ = 243 L.
Ratio = 243/72 = 3.375 = 1.5³. Volume scales with the cube of the linear factor, not the factor itself. The student's claim of 1.5× understates the increase by a large margin; the tank holds 3.375 times as much water.
2. Interpret the surface-area rule and compute the original water surface area to check compliance for 40 cm of fish.
Read the worked answer
The rule "1.5 times the tank's length in cm² per cm of fish" most sensibly means required surface area = 1.5 × length × fish length. For 40 cm of fish and length 60 cm: required = 1.5 × 60 × 40 = 3600 cm².
Actual water surface = 60 × 30 = 1800 cm².
So the tank fails: 1800 < 3600. Note the rule is ambiguous - one could read it as 1.5 × length total, which would be 90 cm² and trivially satisfied. The student should state the interpretation used, because the two readings give opposite verdicts. Under the stricter reading, the tank is inadequate for 40 cm of fish despite holding 72 L.
3. The glass is 0.5 cm thick. Recompute the internal volume and explain why the difference matters more for small tanks.
Read the worked answer
Internal dimensions: (60 − 1) × (30 − 1) × (40 − 1) = 59 × 29 × 39 cm, assuming glass on all faces.
Volume = 59 × 29 × 39 = 66,729 cm³ ≈ 66.73 L.
Difference from the nominal 72 L is about 5.27 L, roughly 7.3% less. The absolute loss is the same order for any tank of this size, but as a proportion it grows for smaller tanks: a 20 × 15 × 20 cm tank loses a much larger fraction. The student should use internal dimensions whenever the litre-per-fish rule is applied, otherwise the tank looks more generous than it is.