Given that a cylinder's radius and height are both reduced to a third of their original size, we need to find the modified surface area formula.
Let's start with the formula for the original surface area of a cylinder:
[tex]\[ \text{Original Surface Area} = 2\pi r (r + h) \][/tex]
If the radius [tex]\(r\)[/tex] and height [tex]\(h\)[/tex] are both shrunk to one-third of their original sizes, the new radius and height are:
[tex]\[ \text{New Radius} = \frac{r}{3} \][/tex]
[tex]\[ \text{New Height} = \frac{h}{3} \][/tex]
Now we substitute these values into the original surface area formula to find the modified surface area:
[tex]\[ \text{Modified Surface Area} = 2\pi \left(\frac{r}{3}\right) \left( \frac{r}{3} + \frac{h}{3} \right) \][/tex]
Simplify inside the parenthesis first:
[tex]\[ \frac{r}{3} + \frac{h}{3} = \frac{r + h}{3} \][/tex]
Now substituting back in:
[tex]\[ \text{Modified Surface Area} = 2\pi \left(\frac{r}{3}\right) \left( \frac{r + h}{3} \right) \][/tex]
Multiply the fractions:
[tex]\[ \text{Modified Surface Area} = 2\pi \cdot \frac{r}{3} \cdot \frac{r + h}{3} = 2\pi \cdot \frac{r (r + h)}{9} \][/tex]
This simplifies to:
[tex]\[ \text{Modified Surface Area} = \frac{2\pi r (r + h)}{9} \][/tex]
This corresponds numerically to:
[tex]\[ \frac{2}{9} \pi r (r + h) \][/tex]
Given this result, we recognize that it equates to approximately:
[tex]\[ \text{Modified Surface Area} \approx 0.698131700797732 \, r \, (h + r) \][/tex]
Therefore, the answer in the provided options which fits the format best is:
[tex]\[ \boxed{\frac{2}{9} \pi \cdot (r^2 + rh)} \][/tex]
Yet, since none of the provided options are in the precisely correct form, but the correct formula is not matching precisely any of the given options, from the provided options, the closest correct form isn't directly listed among A, B, C, or D. The correct modified surface area formula expressed in simplest terms should indeed follow from our calculations in line with options but we should ideally check for any typographic errors or revisit the provided information and context for full clarity.
