To find the angle of elevation from the point where Tyler is standing to the top of the Top Thrill Dragster ride, you can use trigonometry.
Here’s a step-by-step explanation:
1. Understand the problem:
- The Top Thrill Dragster ride is 420 feet tall.
- Tyler is standing 112 feet away from the base of the ride.
2. Form a right triangle:
- The height of the roller coaster forms the opposite side of the right triangle.
- The distance from Tyler to the base of the ride forms the adjacent side of the right triangle.
- The angle of elevation is the angle from the horizontal ground (where Tyler is standing) up to the top of the ride.
3. Use the tangent function:
- The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
- [tex]\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\][/tex]
- In this case:
- Opposite side (height of the ride) = 420 feet
- Adjacent side (distance from the base) = 112 feet
- So, [tex]\[\tan(\theta) = \frac{420}{112}\][/tex]
4. Calculate the angle:
- To find [tex]\(\theta\)[/tex], you need to take the inverse tangent ([tex]\(\arctan\)[/tex]) of the ratio:
- [tex]\[\theta = \arctan\left(\frac{420}{112}\right)\][/tex]
5. Convert the angle from radians to degrees:
- After calculating the inverse tangent, you get an angle in radians. To convert it to degrees, you multiply by [tex]\(\frac{180}{\pi}\)[/tex].
6. Get the numerical value:
- The tangent inverse value results in approximately 1.3102 radians.
- Converting this to degrees: [tex]\[1.3102 \times \frac{180}{\pi} \approx 75.07 \text{ degrees}\][/tex]
Therefore, the angle of elevation from the point where Tyler is standing to the top of the Top Thrill Dragster ride is approximately [tex]\(75.07\)[/tex] degrees.
