To find the width of the river using the angles of elevation given to the tops of poles A and B, we can use trigonometry, specifically the tangent function, which relates the opposite side of a right triangle (the height of the poles in this case) to the adjacent side (the width of the river). We can set up two equations involving the tangents of the angles of elevation and then solve for the width of the river.
Let's denote:
- [tex]\( h \)[/tex] as the height of the poles,
- [tex]\( w \)[/tex] as the width of the river.
From point M directly opposite pole A, we form a right triangle where the angle of elevation to the top of pole A is 22 degrees.
Using the tangent function:
[tex]\[ \text{tan}(22^\circ) = \frac{h}{w} \][/tex]
So, we can express the height of the poles as:
[tex]\[ h = w \cdot \text{tan}(22^\circ) \tag{1} \][/tex]
Similarly, for pole B which is 100m from pole A, from the same point M we form a different right triangle where the angle of elevation to the top of pole B is 19 degrees.
Applying the tangent function here:
[tex]\[ \text{tan}(19^\circ) = \frac{h}{(w + 100)} \][/tex]
Consequently, we can also express the height of the poles in terms of the width of the river and the known distance between the poles:
[tex]\[ h = (w + 100) \cdot \text{tan}(19^\circ) \tag{2} \][/tex]
Since the height of both poles is the same, we can set equation (1) equal to equation (2):
[tex]\[ w \cdot \text{tan}(22^\circ) = (w + 100) \cdot \text{tan}(19^\circ) \][/tex]
Now, let's solve this equation for [tex]\( w \)[/tex], the width of the river:
[tex]\[ w \cdot \text{tan}(22^\circ) = w \cdot \text{tan}(19^\circ) + 100 \cdot \text{tan}(19^\circ) \][/tex]
[tex]\[ w \cdot (\text{tan}(22^\circ) - \text{tan}(19^\circ)) = 100 \cdot \text{tan}(19^\circ) \][/tex]
[tex]\[ w = \frac{100 \cdot \text{tan}(19^\circ)}{(\text{tan}(22^\circ) - \text{tan}(19^\circ))} \][/tex]
Now we have everything in terms of known values and can proceed to calculate the numerical result. We need to find the values of the tangents of 22 and 19 degrees and plug them into the equation we derived.
Since this is a hypothetical explanation, let's carry out the mathematic operation using a calculator or a trigonometry table to find the tangent values:
[tex]\[ \text{tan}(22^\circ) \approx 0.4040 \][/tex]
[tex]\[ \text{tan}(19^\circ) \approx 0.3443 \][/tex]
Using these values:
[tex]\[ w = \frac{100 \cdot 0.3443}{(0.4040 - 0.3443)} \][/tex]
[tex]\[ w = \frac{34.43}{0.0597} \][/tex]
[tex]\[ w \approx 576.88 \][/tex]
Therefore, the width of the river is approximately 576.88 meters.
