Hidden From View
You are snorkeling at the 12-meter-deep water's surface, and there is a flat object one square meter in area suspended horizontally 6 meters above the sea-floor. You are not directly above it. To the nearest cubic meter, how much water does it hide from your view?
The answer is 14.
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1 solution
Relevant wiki: Volume of a Cone
This solution makes use of the fact that the volume of any cone, whatever the shape of its base and however off-center its tip, is one third times the area of its base times its height (the distance from the tip to the plane of the base).
The volume obstructed from view is an oblique conical frustrum---that is, it is a slanty, truncated cone with a not-necessarily-circular base.
The base of this frustrum is the "shadow" your eye casts on the seafloor; it is the same shape as the object but twice as big linearly, and so it has four times the area, namely 4 square meters. Thinking of your eye as the tip of the cone, the entire cone has volume 3 1 ⋅ 4 ⋅ 1 2 = 1 6 . And the cone containing the water from your eye to the object has volume 3 1 ⋅ 1 ⋅ 6 = 2 . And so the volume of the frustrum is 1 4 .