A spherical wooden ball 1 ft in diameter is used as a float for a boat. If the ball sinks to a depth of 7 inches, find the area of the wetted surface.

84 pi

To find the wetted surface area of the spherical wooden ball, we can start by finding the radius of the submerged portion.

The diameter of the wooden ball is given as 1 ft, so the radius (r) is half of the diameter:

r = 1 ft / 2 = 0.5 ft

Now, we need to find the height of the submerged portion, which is given as 7 inches. We convert this to feet:

h = 7 inches * (1 ft / 12 inches) = 7/12 ft

Next, we can use the formula for the wetted surface area of a partially submerged sphere:

A = πr² + 2πrh

Substituting the values we found:

A = π(0.5 ft)² + 2π(0.5 ft)(7/12 ft)

To find the area of the wetted surface, we need to find the surface area of the portion of the sphere that is submerged. Here's how you can go about solving this problem:

1. Determine the radius of the spherical wooden ball: Since the ball has a diameter of 1 ft, the radius is half of the diameter, which is 1/2 ft or 6 inches.

2. Calculate the portion of the sphere that is submerged: Given that the ball sinks to a depth of 7 inches, we know that the portion submerged is 7 inches. Let's denote this as "h" for height.

3. Find the radius of the submerged portion: To determine the radius of the submerged portion, we can use the Pythagorean theorem. The radius of the spherical ball (6 inches) and the height of the submerged portion (7 inches) form a right triangle. Using the Pythagorean theorem, we can solve for the radius of the submerged portion. The formula is:

r_submerged = √(r^2 - h^2)
r_submerged = √(6^2 - 7^2)
r_submerged ≈ √(36 - 49)
r_submerged ≈ √(-13)

Since the square root of a negative number is not a real number, we can conclude that the ball does not sink to a depth of 7 inches. Therefore, there is no wetted surface, and the answer would be zero.