Sure, let's solve this step by step.

1. The given equation is of a circle with radius √81 = 9 (centered at the origin).

2. We are looking for integral solutions, so x and y must be integers.

3. The possible values for x and y range from -9 to 9 inclusive.

4. For each value of x from -9 to 9, the possible values of y are those for which y² ≤ 81 - x².

5. We can count the number of possible y values for each x and sum these up.

6. For x = -9, -8, ..., 8, 9, the number of possible y values are 19, 17, 17, 15, 15, 13, 13, 11, 11, 9, 9, 7, 7, 5, 5, 3, 3, 1, 1 respectively.

7. Summing these up, we get 19*2 + 17*4 + 15*4 + 13*4 + 11*4 + 9*4 + 7*4 + 5*4 + 3*4 + 1*2 = 324.

So, there are 324 integral solutions for x² + y² ≤ 81.

Question

Sure, let's solve this step by step.

1. The given equation is of a circle with radius √81 = 9 (centered at the origin).

2. We are looking for integral solutions, so x and y must be integers.

3. The possible values for x and y range from -9 to 9 inclusive.

4. For each value of x from -9 to 9, the possible values of y are those for which y² ≤ 81 - x².

5. We can count the number of possible y values for each x and sum these up.

6. For x = -9, -8, ..., 8, 9, the number of possible y values are 19, 17, 17, 15, 15, 13, 13, 11, 11, 9, 9, 7, 7, 5, 5, 3, 3, 1, 1 respectively.

7. Summing these up, we get 19*2 + 17*4 + 15*4 + 13*4 + 11*4 + 9*4 + 7*4 + 5*4 + 3*4 + 1*2 = 324.

So, there are 324 integral solutions for x² + y² ≤ 81.

Knowee AI · Accepted Answer