To find out how many times smaller the surface area of a sphere becomes when its radius is multiplied by 1/4, let's start with the formula for the surface area of a sphere, which is:
\[ A = 4 \pi r^2 \]
Now let's consider a sphere with an initial radius \(r_1\). Its surface area would be:
\[ A_1 = 4 \pi r_1^2 \]
If we then multiply the radius by 1/4, we get a new radius \(r_2 = \frac{1}{4} r_1\). The surface area of the sphere with this new radius is:
\[ A_2 = 4 \pi r_2^2 = 4 \pi \left(\frac{1}{4} r_1\right)^2 \]
Now we simplify \(A_2\) by squaring the new radius:
\[ A_2 = 4 \pi \left(\frac{1}{4}^2 r_1^2\right) = 4 \pi \left(\frac{1}{16} r_1^2\right) \]
\[ A_2 = \frac{4 \pi r_1^2}{16} \]
\[ A_2 = \frac{4 \pi r_1^2}{4 \cdot 4} \]
\[ A_2 = \frac{1}{4} \cdot \frac{4 \pi r_1^2}{4} \]
\[ A_2 = \frac{1}{4} \cdot \pi r_1^2 \]
Notice that \(A_2\) is exactly 1/16 of \(A_1\) because when you square (1/4), you get (1/16).
\[ A_2 = \frac{A_1}{16} \]
This means the surface area of the sphere when the radius is multiplied by 1/4 is 16 times smaller than the original surface area.
So the answer is that the surface area of the sphere is 16 times smaller when the radius is multiplied by 1/4.
