Viewing Angle of an Observer While visiting a museum, Marsha Langlois views a pa

Question

Viewing Angle of an Observer While visiting a museum, Marsha Langlois views a painting that is 3 ft high and hangs 6 ft above the ground. See the figure. Assume her eyes are 5 ft above the ground, and let x be the distance from the spot where she is standing to the wall displaying the painting. Show that theta, the viewing angle subtended by the painting, is given by theta = tan^-1(4/x) - tan^-1(1/x). Find the value of x to the nearest hundredth for each value of theta. theta = pi/6 theta = pi/8 Find the value of theta to the nearest hundredth for each value of x. x = 4 x = 3

Explanation / Answer

1) from traingle 1 with base x ft and height 4 ft with angle a

tana = 4/x ; a = tan^-1(4/x)

from traingle 2 with base x ft and height 1 ft with angle b

tanb = 1/x ; b = tan^-1(1/x)

theta = a - b = tan^-1(4/x) - tan^-1(1/x)

2) i) theta = pi/6

pi/6 =tan^-1(4/x) - tan^-1(1/x)

pi/6 = tan^-1[ ( 4/x - 1/x)]/[ 1 +4/x^2]

1/sqrt3 = 3/(x +4/x)

x +4/x = 3/sqrt3 = sqrt3

x^2 - xsqrt3 +4 =0

solve for x

ii) Similarly

theta = pi/8

pi/8 =tan^-1(4/x) - tan^-1(1/x)

pi/8 = tan^-1[ ( 4/x - 1/x)]/[ 1 +4/x^2]

tan(pi/8) = 3/(x +4/x)

0.414(x +4/x) = 3

0.414x^2 + -3x + 1.656 =0

solve for x

c) theta = tan^-1(4/x) - tan^-1(1/x)

plug x = 3 ; theta = tan^-1(4/3) - tan^-1(1/3)

= 53.13 deg - 18.26 deg

= 34.87 deg

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