Final answer:
When the diameter of a sphere is doubled, its radius also doubles, leading to a surface area that is four times larger than the original. Thus, the value of x indicating the increase in surface area is 4.
Step-by-step explanation:
The student asked what happens to the surface area of a sphere if its diameter is doubled. The formula used to calculate the surface area of a sphere is 4πr², where r is the radius of the sphere. Because the diameter is twice as long, the radius will also double. Therefore, if the original radius is r, the new radius is 2r.
Now let's calculate the new surface area with the increased radius:
Surface Area of original sphere = 4πr²
Surface Area of new sphere = 4π(2r)² = 4π(4r²) = 16πr²
Comparing the new surface area to the original, we can see that it is 4 times larger (because 16πr² is 4 times 4πr²). Therefore, the value of x that represents how much greater the surface area becomes when the diameter is doubled is 4. This corresponds to option C.