Question 22

If the value of r is doubled, the corresponding value of S is multiplied by 4.

When the radius \( r \) is doubled in the equation \( S = 4\pi r^2 \), the surface area \( S \) becomes \( S = 4\pi (2r)^2 = 4\pi (4r^2) = 16\pi r^2 \). This shows that \( S \) is multiplied by 4.

A) 1/4

This option suggests that the surface area decreases to a quarter of its original value. However, increasing the radius leads to an increase in surface area, not a decrease, making this option incorrect.

B) 1/2

Choosing this option indicates that the surface area would be half of its initial value. Similar to option A, doubling the radius results in a significant increase in surface area, rendering this option incorrect as well.

C) 2

This choice implies that the surface area would double when the radius is doubled. In reality, the surface area is proportional to the square of the radius; thus, it does not merely double, making this option inaccurate.

D) 4

This option accurately reflects that when the radius \( r \) is doubled, the surface area \( S \) increases by a factor of 4, as derived from the equation \( S = 4\pi (2r)^2 = 16\pi r^2 \), confirming that this is the correct answer.

E) 8

Selecting this option would suggest that the surface area increases by a factor of 8 when the radius is doubled. However, since the relationship between radius and surface area is quadratic, this option is incorrect.

Conclusion

The correct answer is D, as it correctly represents the mathematical relationship between the radius and surface area in the equation provided. All other options fail to recognize the quadratic nature of the equation, leading to miscalculations about how surface area changes with varying radius.