A) To find the area of the dilated rectangle, we first calculate the area of the original rectangle and then apply the scale factor.
The area of the original rectangle is: \[ \text{Area} = \text{Length} \times \text{Width} = 5 , \text{inches} \times 10 , \text{inches} = 50 , \text{square inches}. \]
When the rectangle is dilated by a scale factor of 2.5, the dimensions of the rectangle become: \[ \text{New Length} = 5 \times 2.5 = 12.5 , \text{inches}, \] \[ \text{New Width} = 10 \times 2.5 = 25 , \text{inches}. \]
Now, we calculate the area of the dilated rectangle: \[ \text{Area} = \text{New Length} \times \text{New Width} = 12.5 , \text{inches} \times 25 , \text{inches} = 312.5 , \text{square inches}. \]
So the area of the dilated rectangle is 312.5 square inches.
B) To compare the surface areas of sphere A and sphere B, we first use the formula for the surface area of a sphere: \[ \text{Surface Area} = 4 \pi r^2. \]
For Sphere A with a radius of 4 inches: \[ \text{Surface Area}_A = 4 \pi (4^2) = 4 \pi (16) = 64 \pi , \text{square inches}. \]
For Sphere B with a radius of 10 inches: \[ \text{Surface Area}_B = 4 \pi (10^2) = 4 \pi (100) = 400 \pi , \text{square inches}. \]
Now, we compare their surface areas: The ratio of Sphere B's surface area to Sphere A's surface area is: \[ \frac{\text{Surface Area}_B}{\text{Surface Area}_A} = \frac{400 \pi}{64 \pi} = \frac{400}{64} = 6.25. \]
This means that Sphere B's surface area is 6.25 times as large as Sphere A's.
Therefore, the correct comparison is: Sphere A has a surface area of 64π in² and sphere B has a surface area of 400π in², meaning sphere B's surface area is 6.25 times as large as sphere A's.