Question

The lower edge of a billboard advertisement is 4 ft., the upper edge is 12 ft. above the eye of an observer. At what horizontal distance should he stand if the angle subtended by the billboard is to be greatest?
Group of answer choices​

Answer 1

Given : The lower edge of a billboard advertisement is 4 ft  and the upper edge is 12 ft above the eye of an observer

To Find :  At what horizontal distance should he stand if the angle subtended by the billboard is to be greatest?

Solution:

Let say Man stand at a distance of  x  ft

Angle of Elevation for lower edge of a billboard  = α

Tan α  =  4/x

=>  α  =  Tan ⁻¹(4/x)

Angle of Elevation for upper edge of a billboard  = β

Tan β  =  12/x

=>  β  =  Tan ⁻¹(12/x)

angle subtended by the billboard  = β -   α

Z = β -   α

=> Z = Tan ⁻¹(12/x) -  Tan ⁻¹(4/x)

dZ/dx    =  (1/(1  + (12/x)²) (-12/x²)  -   (1/(1  + (4/x)²) (-4/x²)

=> dZ/dx    =   -12/(x² + 144) + 4/(x² + 16)

dZ/dx    = 0

=>  -12/(x² + 144) + 4/(x² + 16) = 0

=> 1/ (x² + 16)   = 3/(x² + 144)

=> x² + 144 = 3x² + 48

=> 2x² = 96

=> x² = 48

=> x =  4√3

d²Z/dx²    = 24x/(x² + 144)² - 8x/(x² + 16)²

= 24(4√3)/(48 + 144)² - 8(4√3)/(48 + 16)²

= 0.00451 - 0.01353

< 0

Hence  x =  4√3    will give maximum subtended angle

He should stand at a distance of    4√3 m  

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