Question

a cell phone box in the shape of a rectangular prism is shown. the height of the box is 4 cm. original box 4 cm 13.5 cm 18 cm the height of the original box will be increased by 3.5 centimeters so a new instruction manual and an extra battery can be included. which is closest to the total surface area of the new box? a 479 cm² b 707 cm² c 738 cm² d 959 cm²

Explanation:

Step1: Find new height

Original height \( h = 4 \, \text{cm} \), increased by \( 3.5 \, \text{cm} \), so new height \( h_{\text{new}} = 4 + 3.5 = 7.5 \, \text{cm} \).

Step2: Recall surface area formula for rectangular prism

The formula for the surface area \( S \) of a rectangular prism is \( S = 2(lw + lh + wh) \), where \( l = 18 \, \text{cm} \), \( w = 13.5 \, \text{cm} \), \( h = 7.5 \, \text{cm} \).

Step3: Calculate each term

• \( lw = 18 \times 13.5 = 243 \)
• \( lh = 18 \times 7.5 = 135 \)
• \( wh = 13.5 \times 7.5 = 101.25 \)

Step4: Substitute into formula

\( S = 2(243 + 135 + 101.25) = 2(479.25) = 958.5 \approx 959 \, \text{cm}^2 \)? Wait, no, wait—wait, maybe I miscalculated. Wait, no, let's recheck. Wait, original length 18, width 13.5, new height 7.5. Wait, \( lw = 18*13.5 = 243 \), \( lh = 18*7.5 = 135 \), \( wh = 13.5*7.5 = 101.25 \). Sum: 243 + 135 = 378; 378 + 101.25 = 479.25. Multiply by 2: 958.5, which is close to 959. But wait, the options: D is 959. But wait, maybe I made a mistake in height? Wait, original height is 4, increased by 3.5, so 7.5. Correct. Length 18, width 13.5. So surface area: 2(lw + lh + wh) = 2(1813.5 + 187.5 + 13.57.5) = 2(243 + 135 + 101.25) = 2(479.25) = 958.5 ≈ 959. So the answer is D? Wait, but let me check again. Wait, maybe the problem is that I misread the dimensions? Wait, the original box: length 18, width 13.5, height 4. New height 4 + 3.5 = 7.5. So yes, that's correct. So the surface area is 2(1813.5 + 187.5 + 13.57.5) = 2(243 + 135 + 101.25) = 2*479.25 = 958.5, which is closest to 959, so D.

Answer:

D \( 959 \, \text{cm}^2 \)