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Answer :
To find out how many feet the dog must walk to reach the barn door directly below Ted, we can use trigonometry. Here's a step-by-step explanation:
1. Understand the Problem:
- Ted is looking down at his dog from a height of 16 feet.
- The angle of depression from Ted's line of sight to the dog is 40°.
2. Picture the Situation:
- Imagine a right triangle where:
- The vertical side (height) is 16 feet (Ted's eye level above the ground).
- The angle of depression is 40°.
- The horizontal side (distance the dog must walk) is what we're trying to find.
3. Use Trigonometric Ratios:
- The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
- Here, the angle we're dealing with is 40°, the opposite side is the height (16 feet), and the adjacent side is the horizontal distance we want to calculate.
4. Apply the Tangent Formula:
- [tex]\(\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}\)[/tex]
- [tex]\(\tan(40°) = \frac{16}{\text{horizontal distance}}\)[/tex]
5. Solve for the Horizontal Distance:
- Rearrange the formula to find the horizontal distance:
[tex]\[
\text{horizontal distance} = \frac{16}{\tan(40°)}
\][/tex]
6. Calculate the Horizontal Distance:
- Using a calculator, find [tex]\(\tan(40°)\)[/tex] and compute the division:
- [tex]\(\tan(40°) \approx 0.8391\)[/tex]
- [tex]\(\text{horizontal distance} \approx \frac{16}{0.8391} \approx 19.07\)[/tex] feet
7. Round to the Nearest Foot:
- Round 19.07 to the nearest whole number, which gives us 19 feet.
Therefore, the dog must walk approximately 19 feet to reach the barn door directly below Ted.
1. Understand the Problem:
- Ted is looking down at his dog from a height of 16 feet.
- The angle of depression from Ted's line of sight to the dog is 40°.
2. Picture the Situation:
- Imagine a right triangle where:
- The vertical side (height) is 16 feet (Ted's eye level above the ground).
- The angle of depression is 40°.
- The horizontal side (distance the dog must walk) is what we're trying to find.
3. Use Trigonometric Ratios:
- The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
- Here, the angle we're dealing with is 40°, the opposite side is the height (16 feet), and the adjacent side is the horizontal distance we want to calculate.
4. Apply the Tangent Formula:
- [tex]\(\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}\)[/tex]
- [tex]\(\tan(40°) = \frac{16}{\text{horizontal distance}}\)[/tex]
5. Solve for the Horizontal Distance:
- Rearrange the formula to find the horizontal distance:
[tex]\[
\text{horizontal distance} = \frac{16}{\tan(40°)}
\][/tex]
6. Calculate the Horizontal Distance:
- Using a calculator, find [tex]\(\tan(40°)\)[/tex] and compute the division:
- [tex]\(\tan(40°) \approx 0.8391\)[/tex]
- [tex]\(\text{horizontal distance} \approx \frac{16}{0.8391} \approx 19.07\)[/tex] feet
7. Round to the Nearest Foot:
- Round 19.07 to the nearest whole number, which gives us 19 feet.
Therefore, the dog must walk approximately 19 feet to reach the barn door directly below Ted.
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