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We are given that Annabelle and Rich are setting up decorations for their school dance, and we know that Rich is standing 5 feet directly in front of Annabelle under a disco ball. Let y represent the horizontal distance between Rich and the disco ball.
We know that the angle of elevation from Annabelle to the ball is 40∘ and from Rich to the ball is 50∘. Since we are asked to evaluate how high the disco ball is, let x represent this height. Notice that we have two right triangles in our diagram.
(I): LHS⋅y=RHS⋅y
(I): Rearrange equation
(II): x=ytan50∘
(II): LHS⋅(5+y)=RHS⋅(5+y)
(II): Distribute tan40∘
(II): LHS−ytan40∘=RHS−ytan40∘
(II): Factor out y
(II): LHS/(tan50∘−tan40∘)=RHS/(tan50∘−tan40∘)
(II): Rearrange equation
(II): Use a calculator
(II): Round to 1 decimal place(s)
(I): y=11.9
(I): Use a calculator
(I): Round to 1 decimal place(s)