Explanation:

Let’s break it down:

- The volume of the cylindrical tank is pR²(R+7).

- The volume of one hemispherical bowl is (11R³)/42.

- So, the number of bowls, N = [pR²(R+7)] / [(11R³)/42] = [42p(R+7)]/(11R), keeping R as a positive integer.

- This means N has to be a whole number, so 42p(R+7) must be divisible by 11R.

- p, as we use standard values, means 22/7. Substitute that in:

N = [42 × 22/7 × (R+7)] / (11R) = [132(R+7)]/(11R) = 12(R+7)/R.

Now let’s check the options:

- i. N = 13

- 12(R+7)/R = 13 ? 12R+84 = 13R ? R = 84.

- R is an integer. Possible.

- ii. N = 8

- 12(R+7)/R = 8 ? 12R+84 = 8R ? 4R = -84 ? R = -21.

- R can’t be negative. Not allowed.

- iii. N = 16

- 12(R+7)/R = 16 ? 12R+84 = 16R ? 84 = 4R ? R = 21.

- R is positive integer. Possible.

- iv. N = 17

- 12(R+7)/R = 17 ? 12R+84 = 17R ? 84 = 5R ? R = 16.8.

- R is not an integer. Not possible.

So only i and iii work.

Correct Answer:

Option: 3 — Only i and iii are possible values for N.