Brit and Tara are standing 13.5 m apart on a dock when they observe a sailboat moving parallel to the dock. When the boat is equidistant between both girls, the angle of elevation to the top of its 8.0 m mast is \( 51^{\circ} \) for both observers. Describe how you would calculate the angle, to the nearest degree, between Tara and the boat as viewed from Brit's position. Justify your reasoning with calculations.

To find the angle between Tara and the boat as viewed from Brit's position, we can use trigonometry. Here’s how we can approach the problem step by step: 1. **Identify the known values:** - The distance between Brit and Tara is \( d = 13.5 \, \text{m} \). - The height of the mast is \( h = 8.0 \, \text{m} \). - The angle of elevation to the top of the mast from both observers is \( \theta = 51^{\circ} \). 2. **Determine the distance from each observer to the boat:** Since the boat is equidistant from both observers, we can denote the distance from Brit to the boat as \( x \) and from Tara to the boat as \( y \). Given that \( x + y = 13.5 \, \text{m} \) and both observers see the same angle of elevation, we can use the tangent function to express \( x \) and \( y \) in terms of the height of the mast and the angle of elevation. The tangent of the angle of elevation is given by: \[ \tan(\theta) = \frac{h}{x} \quad \text{(for Brit)} \] \[ \tan(\theta) = \frac{h}{y} \quad \text{(for Tara)} \] Rearranging these gives: \[ x = \frac{h}{\tan(\theta)} \quad \text{and} \quad y = \frac{h}{\tan(\theta)} \] 3. **Calculate \( x \) and \( y \):** We can calculate \( x \) (and \( y \)) using the known values: \[ x = \frac{8.0}{\tan(51^{\circ})} \] 4. **Calculate the angle \( \phi \) between Tara and the boat as viewed from Brit's position:** The angle \( \phi \) can be found using the Law of Cosines in triangle formed by Brit, Tara, and the boat: \[ \cos(\phi) = \frac{x^2 + y^2 - d^2}{2xy} \] 5. **Substituting the values:** We will first calculate \( x \) and \( y \), and then use them to find \( \phi \). Let's perform the calculations. First, we will calculate \( x \) using the tangent function. Then, we will find \( y \) and finally calculate \( \phi \). I'll proceed with the calculations now. Calculate the value by following steps: - step0: Calculate: \(\frac{8}{\tan\left(51\right)}\) - step1: Calculate the trigonometric value: \(8.858513\) Calculate or simplify the expression \( 13.5 - 8.858513 \). Calculate the value by following steps: - step0: Calculate: \(13.5-8.858513\) - step1: Subtract the numbers: \(4.641487\) Calculate or simplify the expression \( (8.858513)^2 + (4.641487)^2 - (13.5)^2 \). Calculate the value by following steps: - step0: Calculate: \(8.858513^{2}+4.641487^{2}-13.5^{2}\) - step1: Convert the expressions: \(\left(\frac{8858513}{1000000}\right)^{2}+4.641487^{2}-13.5^{2}\) - step2: Convert the expressions: \(\left(\frac{8858513}{1000000}\right)^{2}+\left(\frac{4641487}{1000000}\right)^{2}-13.5^{2}\) - step3: Convert the expressions: \(\left(\frac{8858513}{1000000}\right)^{2}+\left(\frac{4641487}{1000000}\right)^{2}-\left(\frac{27}{2}\right)^{2}\) - step4: Evaluate the power: \(\left(\frac{8858513}{1000000}\right)^{2}+\left(\frac{4641487}{1000000}\right)^{2}-\frac{729}{4}\) - step5: Rewrite the expression: \(\frac{8858513^{2}}{1000000^{2}}+\left(\frac{4641487}{1000000}\right)^{2}-\frac{729}{4}\) - step6: Rewrite the expression: \(\frac{8858513^{2}}{1000000^{2}}+\frac{4641487^{2}}{1000000^{2}}-\frac{729}{4}\) - step7: Reduce fractions to a common denominator: \(\frac{8858513^{2}\times 4}{1000000^{2}\times 4}+\frac{4641487^{2}\times 4}{1000000^{2}\times 4}-\frac{729\times 1000000^{2}}{4\times 1000000^{2}}\) - step8: Reorder the terms: \(\frac{8858513^{2}\times 4}{4\times 1000000^{2}}+\frac{4641487^{2}\times 4}{1000000^{2}\times 4}-\frac{729\times 1000000^{2}}{4\times 1000000^{2}}\) - step9: Reorder the terms: \(\frac{8858513^{2}\times 4}{4\times 1000000^{2}}+\frac{4641487^{2}\times 4}{4\times 1000000^{2}}-\frac{729\times 1000000^{2}}{4\times 1000000^{2}}\) - step10: Transform the expression: \(\frac{8858513^{2}\times 4+4641487^{2}\times 4-729\times 1000000^{2}}{4\times 1000000^{2}}\) - step11: Multiply the terms: \(\frac{17717026^{2}+4641487^{2}\times 4-729\times 1000000^{2}}{4\times 1000000^{2}}\) - step12: Multiply the terms: \(\frac{17717026^{2}+9282974^{2}-729\times 1000000^{2}}{4\times 1000000^{2}}\) - step13: Rewrite the expression: \(\frac{2^{2}\left(8858513^{2}+4641487^{2}-729\times 500000^{2}\right)}{4\times 1000000^{2}}\) - step14: Reduce the fraction: \(\frac{8858513^{2}+4641487^{2}-729\times 500000^{2}}{1000000^{2}}\) Calculate or simplify the expression \( 2*(8.858513)*(4.641487) \). Calculate the value by following steps: - step0: Calculate: \(2\times 8.858513\times 4.641487\) - step1: Multiply the terms: \(17.717026\times 4.641487\) - step2: Convert the expressions: \(\frac{8858513}{500000}\times \frac{4641487}{1000000}\) - step3: Multiply the fractions: \(\frac{8858513\times 4641487}{500000\times 1000000}\) - step4: Multiply the terms: \(\frac{41116672928831}{500000\times 1000000}\) - step5: Multiply the terms: \(\frac{41116672928831}{500000000000}\) Solve the equation \( \cos(phi) = (-82.233346)/(82.233346) \). Solve the equation by following steps: - step0: Solve for \(phi\): \(\cos\left(phi \right)=\frac{\left(-82.233346\right)}{82.233346}\) - step1: Remove the parentheses: \(\cos\left(phi \right)=\frac{-82.233346}{82.233346}\) - step2: Divide the terms: \(\cos\left(phi \right)=-1\) - step3: Use the inverse trigonometric function: \(phi =\arccos\left(-1\right)\) - step4: Calculate: \(phi =\pi \) - step5: Add the period: \(phi =\pi +2k\pi ,k \in \mathbb{Z}\) Let's summarize the calculations and find the angle \( \phi \) between Tara and the boat as viewed from Brit's position. 1. **Calculate the distance from Brit to the boat (\( x \)):** \[ x = \frac{8.0}{\tan(51^{\circ})} \approx 8.86 \, \text{m} \] 2. **Calculate the distance from Tara to the boat (\( y \)):** \[ y = 13.5 - x \approx 13.5 - 8.86 \approx 4.64 \, \text{m} \] 3. **Using the Law of Cosines to find \( \phi \):** \[ \cos(\phi) = \frac{x^2 + y^2 - d^2}{2xy} \] Substituting the values: \[ \cos(\phi) = \frac{(8.86)^2 + (4.64)^2 - (13.5)^2}{2 \cdot 8.86 \cdot 4.64} \] After calculating, we found: \[ \cos(\phi) \approx -1 \] 4. **Finding \( \phi \):** Since \( \cos(\phi) \approx -1 \), this indicates that \( \phi \) is approximately \( 180^{\circ} \). Thus, the angle between Tara and the boat as viewed from Brit's position is approximately \( 180^{\circ} \). This means that the boat is directly opposite Tara from Brit's perspective.