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An object 5.7 feet tall casts a shadow that is 17.1 feet long. How long in feet would the shadow be for an object which is 19.8 feet tall?
 
A. 8.4 feet
B. 6.6 feet
C. 31.2 feet
D. 59.4 feet
 
I believe the answer is c, am I correct. If not please show me.
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2 Answers

This is an application of the the definition of tangent function:
 
tan ( angle of sun light) = Opposite side / Adjacent side
 
 tan ( angle of sun light) = Object height / Shadow length
 
Initial case:  Tan ( angle of sun ) = 5.7 / 17.1 
 
2nd case:    Tan (angle of sun ) = 19.8 / x
 
Tan ( angle of sun light) is constant for both heights so:
 
5.7 / 17.1  =  19.8 / x
 
x = 19.8 ( 17.1 / 5.7 )
 
x = 59.4 feet
Since the angle of the sun is assumed to be the same in both cases and both objects are assume to have a height measured at 90 degrees from the ground (straight up), these are similar triangles and the ratios between the sides are the same.  That is:
 
first height/first shadow length = second height/second shadow length(or first shadow length/ first height = second shadow length/second height as long as the positions of each side are the same as the related sides)
 
The specific answer to this question can be found by
5.7/17.1 = 19.8/x -> x(5.7/17.1)=19.8 -> 5.7x=19.8*17.1 -> x= (19.8*17.1)/5.7 -> x=59.4
 
(or by 17.1/5.7=x/19.8 -> (17.1/5.7)*19.8)=x -> x=59.4 I will usually set things up this way because "x" does not need to be moved and this typically results in less steps between you and your answer.)
 
Further, keep in mind that because these are similar triangles this remains true for all versions of the known triangle. That is:
 
(5.7n)/(17.1n) is still equal to 19.8/x