Answer :
To solve this question, we'll use the formulas related to the surface area and the volume of a sphere. First, let's find the radius of the sphere using its surface area, and then we will use this radius to calculate the volume.
Step 1: Write down the formula for the surface area (SA) of a sphere:
\[ SA = 4 \pi r^2 \]
Here, SA is the surface area and r is the radius of the sphere.
Step 2: Substitute the given surface area and the value for π (pi) into the formula to solve for r (the radius). We're told to use 3.14 for pi. The surface area given is 2826 square millimeters.
\[ 2826 = 4 \times 3.14 \times r^2 \]
Step 3: Solve the above equation for r^2:
\[ r^2 = \frac{2826}{4 \times 3.14} \]
\[ r^2 = \frac{2826}{12.56} \]
\[ r^2 = 225 \]
Step 4: Find the radius by taking the square root of both sides of the equation:
\[ r = \sqrt{225} \]
\[ r = 15 \text{ mm} \]
Now that we have the radius, we can calculate the volume of the sphere.
Step 5: Write down the formula for the volume (V) of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
Step 6: Substitute the radius and the value for π into the formula to get the volume:
\[ V = \frac{4}{3} \times 3.14 \times 15^3 \]
\[ V = \frac{4}{3} \times 3.14 \times 3375 \]
Step 7: Calculate the volume:
\[ V = 4.1867 \times 3375 \]
\[ V \approx 14137.5 \text{ cubic millimeters} \]
Step 8: Round the volume to the nearest whole number:
\[ V \approx 14138 \text{ mm}^3 \]
Hence, the volume of the sphere, rounded to the nearest whole number, is 14,138 cubic millimeters.
