We can begin by drawing a diagram to help illustrate the problem:
As we are taking into account the opposite and adjacent sides of the triangle, we can use the \(\tan\) trigonometric function to solve for \(x\):
\(\tan52 = \cfrac{x}{12}\)
\((12)(\tan52) = (12)\left(\cfrac{x}{12}\right)\)
\(x = (12)(\tan52)\)
\(x = 15.359 \approx 15.4\;[\text{m}]\)
Therefore, we can determine that the height of the tree to the nearest metre is \(\textbf{15.4 [m]}\).
At the top of a hiking trail, there are 2 vertical posts. One is \(5\;[\text{m}]\) tall and the other is \(7\;[\text{m}]\) tall. The ground between the posts is level, and the bases of the posts are 4m apart. The posts are connected by 2 straight wires.
i. As we have the opposite and adjacent side lengths for both triangles, we can determine the angle each wire makes with the ground using the \(\tan\) trigonometric function:
First, we can determine the angle Wire \(1\) makes with the ground:
\(\text{Wire}_{1}: \tan\theta = \cfrac{5}{4}\)
\(\text{Wire}_{1}: \theta = \tan⁻¹\left(\cfrac{5}{4}\right)\)
\(\text{Wire}_{1}: \theta = 51°\)
Next, we can determine the angle Wire \(2\) makes with the ground:
\(\text{Wire}_{2}: \tan\theta = \cfrac{7}{4}\)
\(\text{Wire}_{2}: \theta = \tan⁻¹\left(\cfrac{7}{4}\right)\)
\(\text{Wire}_{2}: \theta = 60°\)
Therefore, we can determine that Wire's \(\textcolor{red}{\boldsymbol{1}}\) and \(\textcolor{blue}{\boldsymbol{2}}\) make respective angles of \(\boldsymbol{\textcolor{red}{51°}}\) and \(\boldsymbol{\textcolor{blue}{60°}}\) with the ground.
ii. We can determine the lengths of each wire using the Pythagorean Theorem:
First, we can determine the length of the Wire \(1\):
\(\text{Wire}_{1}: c^2 = (5)^2 + (4)^2\)
\(\text{Wire}_{1}: c^2 = 25 + 16\)
\(\text{Wire}_{1}: \sqrt{c^2} = \sqrt{41}\)
\(\text{Wire}_{1}: c = 6.4\;[\text{m}]\)
Next, we can determine the length of the Wire \(2\):
\(\text{Wire}_{2}: c^2 = (7)^2 + (4)^2\)
\(\text{Wire}_{2}: c^2 = 49 + 16\)
\(\text{Wire}_{2}: \sqrt{c^2} = \sqrt{65}\)
\(\text{Wire}_{2}: c = 8.1\;[\text{m}]\)
Therefore, we can determine that the lengths of Wires \(\boldsymbol{\textcolor{red}{1}}\) and \(\boldsymbol{\textcolor{blue}{2}}\) are \(\textcolor{red}{\textbf{6.4 [m]}}\) and \(\textcolor{blue}{\textbf{8.1 [m]}}\) respectively.
First, we can sketch a diagram to help visualize the problem:
As we already have the opposite side length determined, and since we are trying to determine the adjacent side lengths on both sides, we can use the tan trigonometric function to determine the total distance:
First, we can determine how far Aimee is from the pole:
\(\text{Aimee}: \tan52° = \cfrac{8}{x}\)
\(\text{Aimee}: (x)(\tan52°) = (\cancel{x})\left(\cfrac{8}{\cancel{x}}\right)\)
\(\text{Aimee}: \cfrac{\tan52°x}{\tan52°} = \cfrac{8}{\tan52°}\)
\(\text{Aimee}: x = 6.25\;[\text{m}]\)
Next, we can determine how far Russell is from the pole:
\(\text{Russell}: \tan38° = \cfrac{8}{y}\)
\(\text{Russell}: (y)(\tan38°) = (y)\left(\cfrac{8}{y}\right)\)
\(\text{Russell}: \cfrac{\tan38°y}{\tan38°} = \cfrac{8}{\tan38°}\)
\(\text{Russell}: y = 10.24\;[m]\)
Finally, we can determine how far Aimee and Russell are from each other by finding the sum of their respective distances from the pole:
\(\text{Total Distance} = \text{Distance}_{\text{Russell}} + \text{Distance}_{\text{Aimee}}\)
\(\text{Total Distance} = 10.24 + 6.25\)
\(\text{Total Distance} = 16.49\;[\text{m}]\)
Therefore, we can determine that the total distance between Aimee and Russell is \(\textbf{16.49 [m]}\).
Since we have the adjacent side length determined, we need to determine what the individual opposite side lengths are for both triangles. To do this, we can use the tan trigonometric function:
First, we can determine the side length of the shorter building:
\(\text{Tower}_1: \tan28 = \cfrac{x}{50}\)
\(\text{Tower}_1: (50)(\tan28) = (\cancel{50})(\cfrac{x}{\cancel{50}})\)
\(\text{Tower}_1: x = 26.59\;[m]\)
Next, we can determine the side length of the taller building:
\(\text{Tower}_2: \tan48 = \cfrac{y}{50}\)
\(\text{Tower}_2: (50)(\tan48) = (\cancel{50})\left(\cfrac{y}{\cancel{50}}\right)\)
\(\text{Tower}_2: y = 55.53\;[\text{m}]\)
Finally, we can calculate the sum of the \(2\) individual side lengths to determine the total height of the taller building:
\(\text{Total Height} = \text{Tower}_1 + \text{Tower}_2\)
\(\text{Total Height} = 26.59 + 55.53\)
\(\text{Total Height} = 82.12\;[\text{m}]\)
Therefore, we can determine that the height of the taller building is \(\textbf{82.12 [m]}\).