Please help: In PQR, sin P=64, sin R=35, and r=21. Find the length of p.

 I believe there is a mistake in the information provided. The sine of an angle in a triangle cannot be more than 1, since it is defined as the ratio of the length of the side opposite the angle to the length of the triangle's hypotenuse, and the hypotenuse is always the longest side of a right triangle. Therefore, sin P = 64 and sin R = 35 should likely be decimal figures representing sin P = 0.64 and sin R = 0.35 if we're speaking about numerical values within the range of a sine function's output.

Also, the given sine values and the side 'r' doesn't directly indicate the nature of the triangle PQR. If it's a right triangle, we’d also need to know which angle is the right angle to proceed. Nevertheless, let’s proceed under the assumption that the triangle is not a right triangle, the sine values provided are actually 0.64 and 0.35 for P and R, respectively, and that 'r' stands for the radius of the circle circumscribed around the triangle (circumradius).

In such a triangle, the Law of Sines states that the ratio of the length of a side to the sine of the opposite angle is constant for all sides and angles in the triangle:

(a / sin A) = (b / sin B) = (c / sin C)

Where a, b, and c are the lengths of the sides opposite angles A, B, and C, respectively.

Also, the Law of Sines can relate the circumradius (r) to the sides and angles of the triangle:

(a / sin A) = (b / sin B) = (c / sin C) = 2r

Given r = 21, sin P = 0.64, and sin R = 0.35, we want to find the length of side p, opposite angle P.

Using the formula:

p / sin P = 2r

We can plug in the values for r and sin P to solve for p:

p / 0.64 = 2 * 21

p = 0.64 * 2 * 21

p = 0.64 * 42

p ≈ 26.88

Thus, the length of side p is approximately 26.88 units.