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Sagot :
To determine the percentage change in the volume of the cuboid given the changes in its dimensions, let's follow the steps systematically.
1. Initial Dimensions:
- The initial dimensions of the cuboid are [tex]\(x \text{ cm}\)[/tex], [tex]\(x \text{ cm}\)[/tex], and [tex]\(y \text{ cm}\)[/tex].
2. Initial Volume Calculation:
- The initial volume of the cuboid can be calculated using the formula for the volume of a cuboid:
[tex]\[ \text{Volume}_{\text{initial}} = x \cdot x \cdot y = x^2 \cdot y \][/tex]
- Assume initial values of [tex]\(x = 1 \text{ cm}\)[/tex] and [tex]\(y = 1 \text{ cm}\)[/tex] for simplicity.
[tex]\[ \text{Volume}_{\text{initial}} = 1 \cdot 1 \cdot 1 = 1 \text{ cm}^3 \][/tex]
3. Changes in Dimensions:
- [tex]\(x\)[/tex] is increased by [tex]\(20\%\)[/tex]:
[tex]\[ \text{New } x = x \cdot (1 + 0.2) = 1 \cdot 1.2 = 1.2 \text{ cm} \][/tex]
- [tex]\(y\)[/tex] is decreased by [tex]\(10\%\)[/tex]:
[tex]\[ \text{New } y = y \cdot (1 - 0.1) = 1 \cdot 0.9 = 0.9 \text{ cm} \][/tex]
4. New Volume Calculation:
- With the new dimensions, the new volume of the cuboid is:
[tex]\[ \text{Volume}_{\text{new}} = \text{New } x \cdot \text{New } x \cdot \text{New } y = 1.2 \cdot 1.2 \cdot 0.9 \][/tex]
- Simplifying this,
[tex]\[ \text{Volume}_{\text{new}} = 1.44 \cdot 0.9 = 1.296 \text{ cm}^3 \][/tex]
5. Percentage Change in Volume:
- To find the percentage change in volume, we use the formula for percentage change:
[tex]\[ \text{Percentage Change} = \frac{\text{New Volume} - \text{Initial Volume}}{\text{Initial Volume}} \times 100\% \][/tex]
- Substituting the values we calculated,
[tex]\[ \text{Percentage Change} = \frac{1.296 \text{ cm}^3 - 1 \text{ cm}^3}{1 \text{ cm}^3} \times 100\% \][/tex]
[tex]\[ \text{Percentage Change} = \frac{0.296 \text{ cm}^3}{1 \text{ cm}^3} \times 100\% = 29.6\% \][/tex]
Therefore, the percentage change in the volume of the cuboid, given the changes in its dimensions, is a 29.6% increase.
1. Initial Dimensions:
- The initial dimensions of the cuboid are [tex]\(x \text{ cm}\)[/tex], [tex]\(x \text{ cm}\)[/tex], and [tex]\(y \text{ cm}\)[/tex].
2. Initial Volume Calculation:
- The initial volume of the cuboid can be calculated using the formula for the volume of a cuboid:
[tex]\[ \text{Volume}_{\text{initial}} = x \cdot x \cdot y = x^2 \cdot y \][/tex]
- Assume initial values of [tex]\(x = 1 \text{ cm}\)[/tex] and [tex]\(y = 1 \text{ cm}\)[/tex] for simplicity.
[tex]\[ \text{Volume}_{\text{initial}} = 1 \cdot 1 \cdot 1 = 1 \text{ cm}^3 \][/tex]
3. Changes in Dimensions:
- [tex]\(x\)[/tex] is increased by [tex]\(20\%\)[/tex]:
[tex]\[ \text{New } x = x \cdot (1 + 0.2) = 1 \cdot 1.2 = 1.2 \text{ cm} \][/tex]
- [tex]\(y\)[/tex] is decreased by [tex]\(10\%\)[/tex]:
[tex]\[ \text{New } y = y \cdot (1 - 0.1) = 1 \cdot 0.9 = 0.9 \text{ cm} \][/tex]
4. New Volume Calculation:
- With the new dimensions, the new volume of the cuboid is:
[tex]\[ \text{Volume}_{\text{new}} = \text{New } x \cdot \text{New } x \cdot \text{New } y = 1.2 \cdot 1.2 \cdot 0.9 \][/tex]
- Simplifying this,
[tex]\[ \text{Volume}_{\text{new}} = 1.44 \cdot 0.9 = 1.296 \text{ cm}^3 \][/tex]
5. Percentage Change in Volume:
- To find the percentage change in volume, we use the formula for percentage change:
[tex]\[ \text{Percentage Change} = \frac{\text{New Volume} - \text{Initial Volume}}{\text{Initial Volume}} \times 100\% \][/tex]
- Substituting the values we calculated,
[tex]\[ \text{Percentage Change} = \frac{1.296 \text{ cm}^3 - 1 \text{ cm}^3}{1 \text{ cm}^3} \times 100\% \][/tex]
[tex]\[ \text{Percentage Change} = \frac{0.296 \text{ cm}^3}{1 \text{ cm}^3} \times 100\% = 29.6\% \][/tex]
Therefore, the percentage change in the volume of the cuboid, given the changes in its dimensions, is a 29.6% increase.
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