A spherical balloon of radius 120 cm subtends an angle of 60° at the eye of an observer. if the angle of elevation of its centre is 60°, what is the height of the centre of the balloon?
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To calculate the distance the balloon is from the man's eye use the formula
s = r*(theta), where s is the arc length of the portion of a radius r circle subtended
by the angle theta. Then r will be the distance the center of the balloon is from the
man's eye. In this case s = 2 * 60 cm = 120 cm and theta = 60 * pi/180 = pi/3
radians, as theta must be in radians for this formula to be applicable. Thus
r = s / (theta) = 120 / (pi/3) = 360/pi = 114.59 cm to 2 decimal places.
Now if the balloon is at an angle of elevation of 45 degrees, then the height will be the
vertical side of a 45-45-90 triangle with hypotenuse 360/pi cm.. Thus the height of
the center of the balloon will be 360/pi * sin(45) = 360/pi * sqrt(2)/2 = 81.03 cm to 2
decimal places.
Edit: Whoops. I see a problem here, in that the arc length value I should use is not
the diameter of the balloon as the balloon is so close to the man's eye. I'll get
the correct solution to you shortly. :)
Edit #2: O.k., draw a diagram with the man's eye at point P outside a circle of
radius 60 cm., and then draw 2 lines from P to the circle that touch either side
of the circle at points Q and R, i.e., tangent lines. Letting O be the center of the
circle, we see that triangle PQO, (or PRO), is a 30-60-90 triangle with the side OQ
opposite the 30 degree angle being the radius r of the circle and the hypotenuse
PO is the distance from the man's eye to the center of the circle. Thus
sin(30) = OQ / PO = r / PO ---> PO = r / sin(30) = 60 / (1/2) = 120 cm.
So the distance from the man's eye to the center of the balloon is 120 cm. .
Now the balloon's center is at an angle of elevation of 45 degrees, so the
height of the center above the man's eye level will be the vertical side of
a 45-45-90 triangle with hypotenuse 120 cm.. Thus the balloon's center
will be 120 * sin(45) = 120 * sqrt(2)/2 = 84.85 cm above the man's eye level
to 2 decimal places. The actual height of the balloon will be 84.85 cm plus
whatever the height the man's eyes are above ground level.
I left my incorrect attempt at a solution on this post as an example of what
happens when one makes incorrect assumptions. :)
s = r*(theta), where s is the arc length of the portion of a radius r circle subtended
by the angle theta. Then r will be the distance the center of the balloon is from the
man's eye. In this case s = 2 * 60 cm = 120 cm and theta = 60 * pi/180 = pi/3
radians, as theta must be in radians for this formula to be applicable. Thus
r = s / (theta) = 120 / (pi/3) = 360/pi = 114.59 cm to 2 decimal places.
Now if the balloon is at an angle of elevation of 45 degrees, then the height will be the
vertical side of a 45-45-90 triangle with hypotenuse 360/pi cm.. Thus the height of
the center of the balloon will be 360/pi * sin(45) = 360/pi * sqrt(2)/2 = 81.03 cm to 2
decimal places.
Edit: Whoops. I see a problem here, in that the arc length value I should use is not
the diameter of the balloon as the balloon is so close to the man's eye. I'll get
the correct solution to you shortly. :)
Edit #2: O.k., draw a diagram with the man's eye at point P outside a circle of
radius 60 cm., and then draw 2 lines from P to the circle that touch either side
of the circle at points Q and R, i.e., tangent lines. Letting O be the center of the
circle, we see that triangle PQO, (or PRO), is a 30-60-90 triangle with the side OQ
opposite the 30 degree angle being the radius r of the circle and the hypotenuse
PO is the distance from the man's eye to the center of the circle. Thus
sin(30) = OQ / PO = r / PO ---> PO = r / sin(30) = 60 / (1/2) = 120 cm.
So the distance from the man's eye to the center of the balloon is 120 cm. .
Now the balloon's center is at an angle of elevation of 45 degrees, so the
height of the center above the man's eye level will be the vertical side of
a 45-45-90 triangle with hypotenuse 120 cm.. Thus the balloon's center
will be 120 * sin(45) = 120 * sqrt(2)/2 = 84.85 cm above the man's eye level
to 2 decimal places. The actual height of the balloon will be 84.85 cm plus
whatever the height the man's eyes are above ground level.
I left my incorrect attempt at a solution on this post as an example of what
happens when one makes incorrect assumptions. :)
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