Certainly! To find the distance between the man's feet and the top of the lighthouse, we need to think of the problem in terms of right-angled triangles.
Here's the step-by-step solution:
1. Identify the right-angled triangle:
- The base of the triangle is the distance from the man to the base of the lighthouse, which is 8 meters.
- The height of the triangle is the height of the lighthouse, which is 15 meters.
- The hypotenuse of the triangle is the distance between the man's feet and the top of the lighthouse, which we need to find.
2. Apply the Pythagorean theorem:
- The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. It can be written as:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
- Here, [tex]\( a \)[/tex] is the height of the lighthouse, [tex]\( b \)[/tex] is the distance from the man to the base of the lighthouse, and [tex]\( c \)[/tex] is the hypotenuse.
3. Substitute the given values into the theorem:
- Let [tex]\( a = 15 \)[/tex] meters (height of the lighthouse).
- Let [tex]\( b = 8 \)[/tex] meters (distance from the man to the base of the lighthouse).
- We need to find [tex]\( c \)[/tex], the hypotenuse.
4. Calculate the hypotenuse:
- Substitute the values into the Pythagorean theorem:
[tex]\[
15^2 + 8^2 = c^2
\][/tex]
- Calculate the squares of the known sides:
[tex]\[
15^2 = 225
\][/tex]
[tex]\[
8^2 = 64
\][/tex]
- Add these values:
[tex]\[
225 + 64 = 289
\][/tex]
- Now solve for [tex]\( c \)[/tex]:
[tex]\[
c^2 = 289
\][/tex]
- Take the square root of both sides to find [tex]\( c \)[/tex]:
[tex]\[
c = \sqrt{289} = 17
\][/tex]
So, the distance between the man's feet and the top of the lighthouse is 17 meters.
