To find the area of a rectangle, you multiply its length by its width. Given:
- Length = 5,780,000,000 miles
- Width = 4,690,000,000 miles
Let's convert these numbers into scientific notation for easier multiplication.
- Length: \( 5,780,000,000 \) miles = \( 5.78 \times 10^9 \) miles
- Width: \( 4,690,000,000 \) miles = \( 4.69 \times 10^9 \) miles
Now, we can calculate the area:
\[ \text{Area} = \text{Length} \times \text{Width} = (5.78 \times 10^9) \times (4.69 \times 10^9) \]
Using the property of exponents, we multiply the coefficients and add the exponents:
\[ = (5.78 \times 4.69) \times (10^9 \times 10^9) \]
Calculating \( 5.78 \times 4.69 \):
\[ 5.78 \times 4.69 \approx 27.1082 \]
Now for the exponent:
\[ 10^9 \times 10^9 = 10^{9+9} = 10^{18} \]
Putting it all together:
\[ \text{Area} \approx 27.1082 \times 10^{18} \text{ square miles} \]
To express this in standard scientific notation, we can rewrite it:
\[ = 2.71082 \times 10^{19} , \text{square miles} \]
However, since \( 27.1082 \times 10^{18} \) is the result obtained from the multiplication, the answer that fits the format from the options provided would be:
B. \( 27.1082 \times 10^{17} \text{ square miles} \).
Therefore, the correct option is B.