A wooden ball 7 inches in radius sinks to a depth of 10 inches in a river. Find the exposed area of the ball above the water.

just look up the formula for the volume of a spherical cap of radius r and height h. Then plug in your numbers.

Hint: find the volume of the smaller cap and subtract it from the whole ball.

See here:

http://en.wikipedia.org/wiki/Spherical_cap

45 pi

To find the exposed area of the wooden ball above the water, you can start by calculating the volume of the submerged part of the ball:

1. The volume of a sphere can be calculated using the formula: V = (4/3) * π * r^3, where V is the volume and r is the radius.

Given a radius of 7 inches, the volume of the whole ball is:
V_total = (4/3) * π * (7^3) = (4/3) * π * 343 ≈ 1436.75 cubic inches.

2. To calculate the volume of the submerged part, imagine slicing off the submerged section of the ball, creating a hemisphere with a radius of 7 inches and a height of 10 inches.

The volume of a hemisphere can be calculated using the formula: V_hemisphere = (2/3) * π * r^3.

Therefore, the volume of the submerged part is:
V_submerged = (2/3) * π * (7^3) = (2/3) * π * 343 ≈ 904.78 cubic inches.

3. Finally, subtract the volume of the submerged part from the total volume to find the volume of the exposed part above the water level:
V_exposed = V_total - V_submerged = 1436.75 - 904.78 ≈ 531.97 cubic inches.

Since the surface area of a sphere is directly proportional to its volume, the exposed surface area above the water will also be the same proportion of the total surface area.

4. Thus, to find the exposed area of the ball above the water, multiply the surface area of the whole ball by the ratio of the exposed volume to the total volume:

Surface area of a sphere formula: A = 4 * π * r^2.

So, the exposed area above the water is:
A_exposed = 4 * π * (7^2) * (V_exposed / V_total)
= 4 * π * 49 * (531.97 / 1436.75)
≈ 590.55 square inches.

Therefore, the exposed area of the ball above the water is approximately 590.55 square inches.

To find the exposed area of the ball above the water, we need to consider the portion of the ball that is submerged and the portion that is above the water. We can use the concept of similar triangles to calculate the submerged portion.

First, we need to determine the height of the submerged portion of the ball. We know that the radius of the ball is 7 inches and it sinks to a depth of 10 inches. Since the ball is completely submerged, the submerged portion is equal to the diameter of the ball, which is 2 times the radius.

Height of submerged portion = 2 * radius = 2 * 7 = 14 inches

Now, let's consider the two triangles formed by the ball. One triangle is formed by the submerged portion of the ball, and the other triangle is formed by the exposed portion of the ball above the water.

The two triangles are similar since they share the same angles. The ratio of the height of the submerged triangle to the height of the exposed triangle is equal to the ratio of the submerged portion to the exposed portion.

Height of submerged triangle / Height of exposed triangle = Submerged portion / Exposed portion

Using this relationship, we can calculate the height of the exposed portion as follows:

Height of exposed portion = (Height of submerged triangle * Exposed portion) / Submerged portion

Height of exposed portion = (14 * 10) / 14 = 10 inches

Since the height of the exposed portion is equal to the depth of the water in which the ball is submerged, the exposed area of the ball above the water is equal to the area of a circle with a radius of 7 inches.

Area of exposed portion = π * radius^2 = π * 7^2 = 49π square inches

Therefore, the exposed area of the ball above the water is equal to 49π square inches.