To determine the ratio of the volumes of two similar cylinders, you need to know that the volume of a cylinder is proportional to the square of its radius (assuming they have the same height). This means that if you have two cylinders with radii \( r_1 \) and \( r_2 \), the ratio of their volumes \( V_1 \) and \( V_2 \) can be expressed as follows:
[tex]\[ \text{Volume ratio} = \left(\frac{r_1}{r_2}\right)^2 \][/tex]
For the given radii:
[tex]\[ r_1 = 7 \][/tex]
[tex]\[ r_2 = 1 \][/tex]
Now, let's compute the volume ratio step-by-step:
1. Square of the radius of the first cylinder:
[tex]\[ r_1^2 = 7^2 = 49 \][/tex]
2. Square of the radius of the second cylinder:
[tex]\[ r_2^2 = 1^2 = 1 \][/tex]
3. Calculate the ratio of the volumes:
[tex]\[ \text{Volume ratio} = \frac{r_1^2}{r_2^2} = \frac{49}{1} = 49 \][/tex]
Therefore, the ratio of the volumes of the two cylinders is \( \boxed{49:1} \).
Thus, the correct answer is:
D. 49:1.
