Question

The cross-section of a searchlight mirror is shaped like a parabola. The light bulb is located 3 centimeters from the base along the axis of symmetry. If the mirror is 20 centimeters across at the opening, find its depth in centimeters. (Round your answer to the nearest tenth if necessary.)

Answer 1

The depth of the mirror of the cross-section of a searchlight would be 8.33 cm if the light bub has a vertex at 3 cm and the mirror is 20 centimeters across at the origin.

A cross-section is perpendicular to the axis of the symmetry goes through the vertex of the parabola. The cross-sectional shape of the mirrored section of most searchlights or spotlights is parabolic.

• It helps in maximizing the output of light in one direction.

• The equation of the cross-section of the parabola is - $y^{2} = 4ax$, where a is the focus and x is the depth of the mirror from its origin.

Given:

a = 3

y = $\frac{20}{2}$ cm = 10 cm

Solution:

from the equation $y^{2} = 4ax$

$y^{2} = 4*3*x\\ y^{2} = 12x$

putting x, 10 cm in the equation

$x=\frac{10^{2} }{12} \\\\x= \frac{100}{12} \\\\x= 8.33 cm$

thus, the depth of the mirror would be - 8.33 cm

Learn more about other problems of the parabola:

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