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Find the area illuminated by a candle h feet from the surface of a ball R feet in radius. How much surface is illuminated when a candle is 10 ft. away from a ball 5 ft. in radius?
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| Photo by Math Principles in Everyday Life |
Solution:
To understand more the problem, it is better to label further the given figure above as follows
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| Photo by Math Principles in Everyday Life |
A spherical zone or zone, in short is a portion of the surface of a sphere from its circular cross section to its end (for one base) or between two parallel circular planes (for two bases). The above figure is a zone of one base.
In this problem, we will solve for the area of spherical zone of one base. Since we want to solve for the area of the ball that is illuminated by a candle, then we have to solve for the area of the spherical zone of one base at the left side. The area of the spherical zone of one base is given by the formula
where R is the radius of a sphere and H is the height or distance of a circular base to the surface of a sphere. Since H is not given in the problem but the distance of a candle to the ball which is h is given, then we have to use the principles of solving similar triangles in order to solve for the value of H in terms of h and R.
Since the light rays of a candle are tangent to the surface of a ball as shown in the figure, then ∆ABD is a right triangle.
Since the radius of a ball is perpendicular to its surface which is illuminated by a candle as shown in the figure, then ∆BDO is also a right triangle.
∆ABD is similar to ∆BDO because if a rt∆ABO is cut by a line segment BD which is perpendicular to line segment AO, then the two right triangles formed are similar to each other.
Since ∆ABD ~ ∆BDO, then we can solve for the value of H in terms of h, r, and R as follows
Next, we need to eliminate r at the above equation because we want to solve for the value of H in terms of h and R. Apply Pythagorean Theorem for ∆BDO in order to get the value of r, we have
Hence, the above equation becomes
Therefore, the area of the spherical zone is
If h = 10ft. and R = 5 ft., then the area of the ball which is illuminated by a candle is
or
