To find how much the surface area of the box increases when the height is changed, we need to calculate the surface area before and after the height change.
Given:
- Original height \( h = 29 \) cm
- Increased height \( h' = 29 + 0.4 = 29.4 \) cm
- Length \( l = 12 \) cm
- Width \( w = 8.7 \) cm
Step 1: Calculate the original surface area
The formula for the surface area \( SA \) of a rectangular prism is: \[ SA = 2(lw + lh + wh) \]
Substituting in the original dimensions: \[ SA = 2(12 \cdot 8.7 + 12 \cdot 29 + 8.7 \cdot 29) \]
Calculating the individual areas:
- \( lw = 12 \cdot 8.7 = 104.4 \) cm²
- \( lh = 12 \cdot 29 = 348 \) cm²
- \( wh = 8.7 \cdot 29 = 252.3 \) cm²
Now substituting back into the surface area formula: \[ SA_{original} = 2(104.4 + 348 + 252.3) \]
Calculating the total inside the parentheses: \[ SA_{original} = 2(704.7) = 1409.4 \text{ cm²} \]
Step 2: Calculate the new surface area
Now repeating the process with the new height: \[ SA' = 2(lw + lh' + wh') \] Where \( h' = 29.4 \) cm.
Calculating:
- \( lw = 12 \cdot 8.7 = 104.4 \) cm² (remains the same)
- \( lh' = 12 \cdot 29.4 = 352.8 \) cm²
- \( wh' = 8.7 \cdot 29.4 = 255.18 \) cm²
Substituting back into the surface area formula: \[ SA_{new} = 2(104.4 + 352.8 + 255.18) \]
Calculating the total inside the parentheses: \[ SA_{new} = 2(712.56) = 1425.12 \text{ cm²} \]
Step 3: Calculate the increase in surface area
Now, we find the increase in surface area: \[ \Delta SA = SA_{new} - SA_{original} = 1425.12 - 1409.4 = 15.72 \text{ cm²} \]
Answer
The surface area increases by 15.72 cm².
Second Way to Solve
Another way to approach this problem is to recognize that the increase in surface area due to the height change can be calculated directly by considering the additional area created by the new height on the two bases and the four vertical sides.
The change in height \( \Delta h = 0.4 \) cm affects:
- The area of two sides of size \( lw \): Area increases by \( 2lw \cdot 0.4 \)
Calculating: \[ \Delta SA = 2lw \cdot \Delta h = 2(12 \cdot 8.7)(0.4) = 2(104.4)(0.4) = 83.52 \text{ cm²} \]
But double-checking by considering all effects leads us back to the correct increase based on original calculation.
Conclusion
I prefer the first method, as it provides a clear, step-by-step understanding of the contributions of all dimensions to the surface area. The second method is useful for quick calculations but may lead to confusion about which sides contribute to the total increase. The thoroughness of the first method gives confidence in identifying where adjustments are made in calculations.