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.
the area of a spherical cap of height h on a sphere of radius r is
A = 2 pi R h
= 2 pi * 7 * 4
= 56 pi = 175.9 in^2
58 cm times 55.6. try to find the area for this problem
To find the exposed area of the ball above the water, we need to calculate the surface area of the spherical shape that is above the water level.
First, let's determine the total volume of the ball. The formula for the volume of a sphere is:
V = (4/3) * π * r³
where V is the volume and r is the radius.
In this case, the radius is given as 7 inches. Thus:
V = (4/3) * π * 7³
V = (4/3) * π * 343
V = 4,546.67 cubic inches
Now, we need to find the volume of the water displaced by the submerged part of the ball. The volume of the water displaced is equal to the volume of the submerged portion of the ball.
The formula for the volume of a cylinder is:
V = π * r² * h
where V is the volume, r is the radius, and h is the height.
In this case, the radius is 7 inches and the height is the depth to which the ball sinks, which is 10 inches. Thus:
V = π * 7² * 10
V = π * 49 * 10
V = 490π cubic inches
The volume of the exposed part of the ball is the difference between the total volume and the volume of water displaced:
Exposure Volume = Total Volume - Displacement Volume
Exposure Volume = 4,546.67 - 490π
Exposure Volume ≈ 10,229.42 cubic inches
Finally, we need to calculate the exposed surface area of the ball. The formula for the surface area of a sphere is:
A = 4 * π * r²
where A is the surface area and r is the radius.
In this case, the radius is 7 inches. Thus:
A = 4 * π * 7²
A = 4 * π * 49
A = 196π square inches
Now, we can find the exposed area above the water level. Using the ratio of the exposed volume to the total volume, we can calculate the exposed area:
Exposed Area = (Exposure Volume / Total Volume) * Surface Area
Exposed Area = (10,229.42 / 4,546.67) * 196π
Exposed Area ≈ 436.96 square inches
Therefore, the exposed area of the wooden ball above the water is approximately 436.96 square inches.