The surface area of a rectangular solid measuring `5times6times8` is how much greater than the surface area of a rectangular solid measuring `3times6times8`?

Options

A. `12`
B. `24`
C. `48`
D. `56`

Answer

To find how much greater the surface area of a rectangular solid measuring \(5 \times 6 \times 8\) is than that of a rectangular solid measuring \(3 \times 6 \times 8\), we will follow these steps:

### Step 1: Calculate the surface area of the first rectangular solid
The formula for the surface area \(SA\) of a rectangular solid is given by:
\[ SA = 2 \times (lw + lh + wh) \]
where \(l\) is the length, \(w\) is the width, and \(h\) is the height.

For the first solid with dimensions \(5\), \(6\), and \(8\):
- Length \(l = 5\)
- Width \(w = 6\)
- Height \(h = 8\)

Substituting the values into the formula:
\[ SA_1 = 2 \times (5 \times 6 + 5 \times 8 + 6 \times 8) \]

### Step 2: Calculate each term inside the parentheses
Calculating each term:
- \(5 \times 6 = 30\)
- \(5 \times 8 = 40\)
- \(6 \times 8 = 48\)

Now, add these values:
\[ 30 + 40 + 48 = 118 \]

### Step 3: Multiply by 2 to find the total surface area
Now, multiply by 2 to find the surface area:
\[ SA_1 = 2 \times 118 = 236 \text{ square units} \]

### Step 4: Calculate the surface area of the second rectangular solid
For the second solid with dimensions \(3\), \(6\), and \(8\):
- Length \(l = 3\)
- Width \(w = 6\)
- Height \(h = 8\)

Using the same formula:
\[ SA_2 = 2 \times (3 \times 6 + 3 \times 8 + 6 \times 8) \]

### Step 5: Calculate each term inside the parentheses for the second solid
Calculating each term:
- \(3 \times 6 = 18\)
- \(3 \times 8 = 24\)
- \(6 \times 8 = 48\)

Now, add these values:
\[ 18 + 24 + 48 = 90 \]

### Step 6: Multiply by 2 to find the total surface area of the second solid
Now, multiply by 2 to find the surface area:
\[ SA_2 = 2 \times 90 = 180 \text{ square units} \]

### Step 7: Find the difference in surface areas
Now, we need to find how much greater the surface area of the first solid is than the second solid:
\[ \text{Difference} = SA_1 - SA_2 = 236 - 180 \]

### Step 8: Calculate the difference
Calculating the difference:
\[ \text{Difference} = 56 \text{ square units} \]

### Final Answer
The surface area of the rectangular solid measuring \(5 \times 6 \times 8\) is \(56\) square units greater than that of the rectangular solid measuring \(3 \times 6 \times 8\).

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