Question

question 3 (2 points)
an airplane climbs at an angle of 18 degrees with the ground. find the ground distance the plane travels as it moves 2500 m through the air. give your answer to the nearest 100 meters.

meters
blank 1:

question 4 (2 points)
a guy wire reaches from the top of a 120 m television transmitter tower to ground. the wire makes a 63 degree angle with the ground. find the length of the guy wire to the nearest meter.

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Trigonometric Ratios"
],
"new_concepts": [],
"current_concepts": [
"Trigonometric Ratios",
"Right Triangle Trigonometry"
]
}
</pre_analysis>

<reasoning>

Question 3: Set up the trigonometric equation

\[

$$\begin{aligned} &\cos(18^\circ) = \frac{x}{2500} \end{aligned}$$

\]

Question 3: Solve for x and round to the nearest 100 meters

\[

$$\begin{aligned} &x = 2500 \cdot \cos(18^\circ) \\ &x \approx 2500 \cdot 0.9510565 \approx 2377.64 \text{ m} \\ &x \approx 2400 \text{ m} \end{aligned}$$

\]

Question 4: Set up the trigonometric equation and solve for x

\[

$$\begin{aligned} &\sin(63^\circ) = \frac{120}{x} \\ &x = \frac{120}{\sin(63^\circ)} \\ &x \approx \frac{120}{0.8910065} \approx 134.68 \text{ m} \\ &x \approx 135 \text{ m} \end{aligned}$$

\]
</reasoning>

<answer>

Question 3

An airplane climbs at an angle of 18 degrees with the ground. Find the ground distance the plane travels as it moves 2500 m through the air. Give your answer to the nearest 100 meters.

<blank>2400</blank> meters

Question 4

A guy wire reaches from the top of a 120 m television transmitter tower to ground. The wire makes a 63 degree angle with the ground. Find the length of the guy wire to the nearest meter.

<blank>135</blank> meters
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Trigonometric Ratios"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Trigonometric Ratios"
],
"new_concepts": [],
"current_concepts": [
"Trigonometric Ratios",
"Right Triangle Trigonometry"
]
}
</pre_analysis>

<reasoning>

Question 3: Set up the trigonometric equation

\[

$$\begin{aligned} &\cos(18^\circ) = \frac{x}{2500} \end{aligned}$$

\]

Question 3: Solve for x and round to the nearest 100 meters

\[

$$\begin{aligned} &x = 2500 \cdot \cos(18^\circ) \\ &x \approx 2500 \cdot 0.9510565 \approx 2377.64 \text{ m} \\ &x \approx 2400 \text{ m} \end{aligned}$$

\]

Question 4: Set up the trigonometric equation and solve for x

\[

$$\begin{aligned} &\sin(63^\circ) = \frac{120}{x} \\ &x = \frac{120}{\sin(63^\circ)} \\ &x \approx \frac{120}{0.8910065} \approx 134.68 \text{ m} \\ &x \approx 135 \text{ m} \end{aligned}$$

\]
</reasoning>

<answer>

Question 3

An airplane climbs at an angle of 18 degrees with the ground. Find the ground distance the plane travels as it moves 2500 m through the air. Give your answer to the nearest 100 meters.

<blank>2400</blank> meters

Question 4

A guy wire reaches from the top of a 120 m television transmitter tower to ground. The wire makes a 63 degree angle with the ground. Find the length of the guy wire to the nearest meter.

<blank>135</blank> meters
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Trigonometric Ratios"
]
}
</post_analysis>