3D Geometry: Comparing Surface Areas of 70×25×12 vs 82×7×13 Solar Panel Cuboids

To solve this problem, we'll calculate the surface area of each cuboid solar panel and compare them:

• Step 1: Identify dimensions for prototype A: length $l = 70$, width $w = 25$, height $h = 12$.
• Step 2: Calculate the surface area of A using the formula $2(lw + lh + wh)$.
  Surface area $= 2(70 \times 25 + 70 \times 12 + 25 \times 12)$.
• Step 3: Perform the calculations:
  $= 2(1750 + 840 + 300)$
  $= 2(2890)$
  $= 5780$.
• Step 4: Identify dimensions for prototype B: length $l = 82$, width $w = 7$, height $h = 13$.
• Step 5: Calculate the surface area of B using the same formula.
  Surface area $= 2(82 \times 7 + 82 \times 13 + 7 \times 13)$.
• Step 6: Perform the calculations:
  $= 2(574 + 1066 + 91)$
  $= 2(1731)$
  $= 3462$.
• Step 7: Compare the surface areas: $5780$ (A) vs. $3462$ (B).

Therefore, prototype A will absorb more of the sun's energy because it has a larger total surface area.

Thus, the solution to the problem is A.