Katie is sailing close to the coast. She measured the angle of elevation to the
top of the lighthouse; it was 0° 24’. Calculate her distance from the lighthouse.
Note: The lighthouse is closer than the horizon.
To calculate Katie's distance from the lighthouse, we can use trigonometry and the concept of angle of elevation.
First, let's clarify the given information:
- The angle of elevation to the top of the lighthouse is 0° 24'.
Now, let's understand the concept of angle of elevation:
The angle of elevation is the angle between the horizontal line of sight and the inclined line of sight from an observer to an object above the horizontal line. In this case, the horizontal line is the line parallel to the coast, and the inclined line of sight is from Katie's position to the top of the lighthouse.
To solve this problem, we can use the tangent function, as we have the angle of elevation and we want to find the distance.
The tangent of an angle is the ratio of the opposite side to the adjacent side of a right triangle. In this case, the angle of elevation is the angle between the line from Katie to the top of the lighthouse and the horizontal line along the coast.
Let's break down the given angle of elevation:
- 0° 24' can be converted to decimal form by dividing the number of minutes (24) by 60.
So, 0° 24' = 0 + (24/60) = 0.4 degrees.
Now, we can use the tangent function to find the distance:
tan(angle of elevation) = opposite side / adjacent side
Let's assume the distance from Katie to the lighthouse is represented by 'd':
tan(0.4 degrees) = height of the lighthouse / d
Since the height of the lighthouse is not given, we can't directly solve for 'd'. However, we can use other information to find the height and then calculate 'd'.
If we know the height of the lighthouse, we can use the tangent equation to find the distance. For example, if we know the height of the lighthouse is 50 meters:
tan(0.4 degrees) = 50 / d
Now, we can rearrange the equation to solve for 'd':
d = 50 / tan(0.4 degrees)
Using a scientific calculator, we can find the value of tan(0.4 degrees) and then calculate 'd' using the given equation.
Please note that the final calculated value of 'd' will be in the same unit as the height of the lighthouse; in this case, it would be meters.
So, to calculate Katie's distance from the lighthouse, we need the height of the lighthouse.