Explain how an 8" x 8" x 16" block can make 8 square feet of wall.

Steps to Solve

1. Convert block dimensions to feet We start by converting the dimensions of the block from inches to feet. Since there are 12 inches in a foot, we convert each dimension by dividing by 12.

For the block:

• Length = $ \frac{8}{12} = \frac{2}{3} \text{ feet} $
• Width = $ \frac{8}{12} = \frac{2}{3} \text{ feet} $
• Height = $ \frac{16}{12} = \frac{4}{3} \text{ feet} $

2. Calculate the block's surface area Next, we compute the surface area of one block. The surface area (SA) of a rectangular block can be calculated with the formula:

$$ SA = 2(lw + lh + wh) $$

Substituting the dimensions we calculated: $$ SA = 2\left(\frac{2}{3} \cdot \frac{2}{3} + \frac{2}{3} \cdot \frac{4}{3} + \frac{2}{3} \cdot \frac{4}{3}\right) $$

3. Compute the numerical value for the surface area We can simplify the calculation:

First, calculate each term:

• $lw = \frac{2}{3} \cdot \frac{2}{3} = \frac{4}{9}$
• $lh = \frac{2}{3} \cdot \frac{4}{3} = \frac{8}{9}$
• $wh = \frac{2}{3} \cdot \frac{4}{3} = \frac{8}{9}$

Add these together: $$ lw + lh + wh = \frac{4}{9} + \frac{8}{9} + \frac{8}{9} = \frac{20}{9} $$

Now, plug this back into the surface area formula: $$ SA = 2 \left(\frac{20}{9}\right) = \frac{40}{9} \text{ square feet} $$

4. Determine how many blocks are needed for the wall To find out how many blocks are needed for a wall with an area of 8 square feet, we divide the total area of the wall by the surface area of one block:

$$ \text{Number of blocks} = \frac{\text{Area of wall}}{\text{Area of one block}} = \frac{8}{\frac{40}{9}} = 8 \cdot \frac{9}{40} = \frac{72}{40} = 1.8 $$

Since we cannot use a fraction of a block, we will round up to the nearest whole number, which gives us 2.