Question

vic has some blocks that are right rectangular prisms with the measurements shown.
vic stacks the blocks to make a tower, keeping the largest faces of each block on the top and bottom.
how many blocks are stacked to create a tower with a surface area between 260 square centimeters and 270 square centimeters?
calculator
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Explanation:

Step1: Find the largest face area

The faces of the rectangular prism have areas: \(10\times3 = 30\), \(10\times2 = 20\), \(3\times2 = 6\). The largest face is \(10\times3 = 30\) \(cm^2\).

Step2: Surface area of one block

The surface area of a rectangular prism is \(2(lw + lh + wh)\). For one block: \(l = 10\), \(w = 3\), \(h = 2\). So surface area \(S_1=2(10\times3 + 10\times2 + 3\times2)=2(30 + 20 + 6)=2\times56 = 112\) \(cm^2\).

Step3: Surface area when stacking \(n\) blocks

When stacking \(n\) blocks, the top and bottom faces remain as the largest faces (\(30\) each), and the lateral faces: the perimeter of the largest face times the total height. The height of one block is \(2\) cm (since we stack with largest faces top/bottom, the height per block is \(2\)). So total height \(h = 2n\). The lateral surface area is perimeter of largest face (\(2(10 + 3)=26\)) times height (\(2n\)): \(26\times2n = 52n\). The total surface area \(S = 2\times30+52n=60 + 52n\).

Step4: Solve for \(n\)

We need \(260 < 60 + 52n < 270\). Subtract 60: \(200 < 52n < 210\). Divide by 52: \(\frac{200}{52}\approx3.85 < n < \frac{210}{52}\approx4.04\). Since \(n\) must be an integer, \(n = 4\).

Answer:

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