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balu736 [363]
3 years ago
6

A flagpole casts a shadow that is 23.5 feet long. At the same time of day, mrs gilbert who is 5.5 feet tall casts a shadow that

is 7.5 feet long. How tall in feet is the flagpole?
Mathematics
1 answer:
madreJ [45]3 years ago
8 0
Shadow of the flagpole=23.5f
Sf=23.5
shadow of gilbert=7.5f
Sg=7.5
height of gilbert=5.5f
hg=5.5
Sf/hf=Sg/hg
23.5/hf=7.5/5.5
23.5*5.5/7.5=hf
(47/2*11/2)/7.5=hf
(517/4)/(30/4)=hf
(517*4)/(4*30)=hf
517/30=hf
17+7/30=hf
17.233.. feet=height of flagpole
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Two lighthouses are located 75 miles from one another on a north-south line. If a boat is spotted S 40o E from the northern ligh
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Refer to the diagram attached. Denote the northern lighthouse as \rm N, the southern lighthouse as \rm S, and the boat as \rm B. These three points would form a triangle.

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By the law of sine, the length of a side in a given triangle would be proportional to the angle opposite to that side. For example, in the triangle in this question, \angle {\rm B} is opposite to side \rm NS, whereas \angle {\rm S} is opposite to side {\rm NB}. Therefore:

\begin{aligned} \frac{\text{length of NS}}{\sin(\angle {\rm B})} = \frac{\text{length of NB}}{\sin(\angle {\rm S})} \end{aligned}.

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\begin{aligned} \frac{75\; \rm mi}{\sin(119^{\circ})} = \frac{\text{length of NB}}{\sin(21^{\circ})} \end{aligned}.

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\begin{aligned} & \text{length of NB} \\ =\; & (75\; \rm mi) \times \frac{\sin(21^{\circ})}{\sin(119^{\circ})} \\ \approx\; & 30.73\; \rm mi\end{aligned}.

(Round to at least one more decimal places than the values in the choices.)

Likewise, with \angle {\rm N} is opposite to side {\rm SB}, the following would also hold:

\begin{aligned} \frac{\text{length of NS}}{\sin(\angle {\rm B})} = \frac{\text{length of SB}}{\sin(\angle {\rm N})} \end{aligned}.

\begin{aligned} \frac{75\; \rm mi}{\sin(119^{\circ})} = \frac{\text{length of SB}}{\sin(40^{\circ})} \end{aligned}.

\begin{aligned} & \text{length of SB} \\ =\; & (75\; \rm mi) \times \frac{\sin(40^{\circ})}{\sin(119^{\circ})} \\ \approx\; & 55.12\; \rm mi\end{aligned}.

In other words, the distance between the northern lighthouse and the boat is approximately 30.73\; \rm mi, whereas the distance between the southern lighthouse and the boat is approximately 55.12\; \rm mi. Hence the conclusion.

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