Question

To estimate the height of a stone figure Anya holds a small square up to her eyes and walks backwards from the figure she stops when the bottom of the figure aligns with the bottom edge of the square and the top of the figure aligns with the top edge of the square her eye level is 1.84m from the ground she is 3.50m from the figure what is the height of the figure to the nearest hundredth of a meter

Answer 1

SOLUTION

Let us make a sketch that will help us solve the question

From the image I provided, let the Height of the figure be H.

Then this means that the height of the figure H is

$H=x+1.84$

Note that I have splitted the triangle into two.

Now, we need to find the two acute angles

$\begin{gathered} \alpha\text{ and }\theta\text{ in the bigger triangle.} \\ \text{Note that }\alpha+\theta+90^o=180^o\text{ (sum of angles in the big triangle)} \end{gathered}$

Now, from the triangles splitted in the right side, both are right-angles just like the first (biggest) triangle.

So we will use the trig-ratio, SOHCAHTOA to find the acute angles.

From the smallest triangle in the image,

$\begin{gathered} \tan \alpha=\frac{opposite}{\text{adjacent }} \\ \tan \alpha=\frac{3.5}{\text{1.84 }} \\ \alpha=\tan ^{-1}\frac{3.5}{\text{1.84 }} \\ \alpha=62.27^o \end{gathered}$

Now, the second acute angle becomes

$\begin{gathered} \text{ }\alpha+\theta+90^o=180^o \\ 62.27+\theta+90=180 \\ \theta+152.27=180 \\ \theta=180-152.27 \\ \theta=27.73^o \end{gathered}$

The other triangle that is the bigger one which was splitted has opposite side of 3.5m and adjacent side x, hence

$\begin{gathered} \tan \theta=\frac{opposite}{\text{adjacent}} \\ \tan 27.73=\frac{3.5}{x} \\ x\times\tan 27.73=3.5 \\ x=\frac{3.5}{\tan 27.73} \\ x=6.65804m \end{gathered}$

Now H becomes

$\begin{gathered} H=x+1.84 \\ H=6.65804+1.84 \\ H=8.49804 \end{gathered}$

Hence, the height of the figure is 8.50 m to the nearest hundreth