Question

a. what is the measure of the shortest side of the sail on the model? round your answer to the nearest tenth.
b. what is the measure of the longest side? round your answer to the nearest tenth.

1. what is the area of the given triangle? round to the nearest tenth. (1 point)
2. a local park is located on a triangular piece of land. what is the area of the park to the nearest tenth? (1 point)

Explanation:

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</pre_analysis>

<reasoning>

Calculate the shortest and longest sides of the sail

\[

$$\begin{aligned} &\triangle EFG \text{ is a } 30^\circ\text{-}60^\circ\text{-}90^\circ \text{ triangle with hypotenuse } EF = 28\text{ in.} \\ &\text{Shortest side } EG = EF \cdot \cos(60^\circ) = 28 \cdot 0.5 = 14.0\text{ in.} \\ &\text{Longest leg } FG = EF \cdot \sin(60^\circ) = 28 \cdot \frac{\sqrt{3}}{2} \approx 24.2487\text{ in.} \approx 24.2\text{ in.} \end{aligned}$$

\]

Calculate the area of triangle ABC

\[

$$\begin{aligned} &\text{Area} = \frac{1}{2} \cdot BC \cdot AC \cdot \sin(C) \\ &\text{Area} = \frac{1}{2} \cdot 13 \cdot 7 \cdot \sin(38^\circ) \approx 45.5 \cdot 0.61566 \approx 28.0126\text{ cm}^2 \approx 28.0\text{ cm}^2 \end{aligned}$$

\]

Calculate the area of the park triangle RST

\[

$$\begin{aligned} &\text{Area} = \frac{1}{2} \cdot RS \cdot RT \cdot \sin(R) \\ &\text{Area} = \frac{1}{2} \cdot 75 \cdot 150 \cdot \sin(40^\circ) \approx 5625 \cdot 0.64279 \approx 3615.68\text{ m}^2 \approx 3615.7\text{ m}^2 \end{aligned}$$

\]
</reasoning>

<answer>

Question a

\(14.0\text{ in.}\)

Question b

\(24.2\text{ in.}\)

Question 9

\(28.0\text{ cm}^2\)

Question 10

\(3615.7\text{ m}^2\)
</answer>

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"Mathematics",
"Geometry",
"Trigonometric Ratios"
]
}
</post_analysis>

Answer:

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<reasoning>

Calculate the shortest and longest sides of the sail

\[

$$\begin{aligned} &\triangle EFG \text{ is a } 30^\circ\text{-}60^\circ\text{-}90^\circ \text{ triangle with hypotenuse } EF = 28\text{ in.} \\ &\text{Shortest side } EG = EF \cdot \cos(60^\circ) = 28 \cdot 0.5 = 14.0\text{ in.} \\ &\text{Longest leg } FG = EF \cdot \sin(60^\circ) = 28 \cdot \frac{\sqrt{3}}{2} \approx 24.2487\text{ in.} \approx 24.2\text{ in.} \end{aligned}$$

\]

Calculate the area of triangle ABC

\[

$$\begin{aligned} &\text{Area} = \frac{1}{2} \cdot BC \cdot AC \cdot \sin(C) \\ &\text{Area} = \frac{1}{2} \cdot 13 \cdot 7 \cdot \sin(38^\circ) \approx 45.5 \cdot 0.61566 \approx 28.0126\text{ cm}^2 \approx 28.0\text{ cm}^2 \end{aligned}$$

\]

Calculate the area of the park triangle RST

\[

$$\begin{aligned} &\text{Area} = \frac{1}{2} \cdot RS \cdot RT \cdot \sin(R) \\ &\text{Area} = \frac{1}{2} \cdot 75 \cdot 150 \cdot \sin(40^\circ) \approx 5625 \cdot 0.64279 \approx 3615.68\text{ m}^2 \approx 3615.7\text{ m}^2 \end{aligned}$$

\]
</reasoning>

<answer>

Question a

\(14.0\text{ in.}\)

Question b

\(24.2\text{ in.}\)

Question 9

\(28.0\text{ cm}^2\)

Question 10

\(3615.7\text{ m}^2\)
</answer>

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