To determine where the light is concentrated by a mirror shaped like a paraboloid of revolution, we need to calculate the focal length of the mirror. The focal length [tex]\( f \)[/tex] for a paraboloid of revolution is given by the formula:
[tex]\[ f = \frac{d^2}{16D} \][/tex]
where:
- [tex]\( d \)[/tex] is the diameter of the mirror (across the opening),
- [tex]\( D \)[/tex] is the depth of the mirror (from the vertex to the deepest point).
Let's use the given values:
- The diameter [tex]\( d \)[/tex] of the mirror is 20 inches.
- The depth [tex]\( D \)[/tex] of the mirror is 5 feet, which we need to convert to inches. Since 1 foot equals 12 inches, 5 feet is:
[tex]\[ 5 \text{ feet} = 5 \times 12 \text{ inches} = 60 \text{ inches} \][/tex]
Now, substitute the values into the focal length formula:
[tex]\[ f = \frac{d^2}{16D} = \frac{(20 \text{ inches})^2}{16 \times 60 \text{ inches}} \][/tex]
[tex]\[ f = \frac{400 \text{ square inches}}{960 \text{ inches}} \][/tex]
[tex]\[ f = \frac{400}{960} \][/tex]
[tex]\[ f = \frac{5}{12} \][/tex]
[tex]\[ f \approx 0.4166666666666667 \text{ inches} \][/tex]
Therefore, the light will be concentrated approximately 0.4 inches from the vertex of the mirror.
The correct answer is:
- 0.4 inches from the vertex
