Answer:
Denote the height of the can corresponding to the 150 cm² label as h₁ and the height of the can corresponding to the 400 cm² label as h₂.
We know that the area of a label is equal to the circumference of the can multiplied by its height.
For the 250 cm² label:
Area = 250 cm²
Circumference = 250 cm² / 8 cm = 31.25 cm (since circumference = Area / height)
Height = 8 cm (given)
For the 150 cm² label:
Area = 150 cm²
Circumference = 150 cm² / h₁
Height = h₁ (to be determined)
For the 400 cm² label:
Area = 400 cm²
Circumference = 400 cm² / h₂
Height = h₂ (to be determined)
Since the labels are similar in shape, the ratios of their corresponding measurements (heights and circumferences) will be the same.
Setting up the proportions:
250 cm² / 8 cm = 150 cm² / h₁ = 400 cm² / h₂
To find h₁, we can solve the second ratio:
150 cm² / h₁ = 250 cm² / 8 cm
Cross-multiplying:
150 cm² * 8 cm = 250 cm² * h₁
1200 cm² = 250 cm² * h₁
Dividing both sides by 250 cm²:
1200 cm² / 250 cm² = h₁
h₁ ≈ 4.8 cm
Therefore, the height of the can that the 150 cm² label would fit around is approximately 4.8 cm.
To find h₂, we can solve the third ratio:
400 cm² / h₂ = 250 cm² / 8 cm
Cross-multiplying:
400 cm² * 8 cm = 250 cm² * h₂
3200 cm² = 250 cm² * h₂
Dividing both sides by 250 cm²:
3200 cm² / 250 cm² = h₂
h₂ ≈ 12.8 cm
The height of the can that the 400 cm² label would fit around is approximately 12.8 cm.