Sure, let's solve this step by step with the given information.
1. Understand the Problem:
- You have a right triangle formed by the ramp, the height of the truck bed, and the distance from the door to the bottom of the ramp.
- The length of the ramp (the hypotenuse of the triangle) is 8 feet.
- The height of the truck bed (one leg of the triangle) is 2 feet.
- We need to find the distance from the door to the truck (the other leg of the triangle).
2. Use the Pythagorean Theorem:
- The Pythagorean Theorem states that in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse.
- Mathematically, this is represented as:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs.
3. Assign the Variables:
- Let \( c = 8 \) feet (the length of the ramp).
- Let \( a = 2 \) feet (the height of the truck bed).
- Let \( b \) be the distance from the door to the truck that we need to find.
4. Rearrange the Pythagorean Theorem to Solve for \( b \):
[tex]\[
b^2 = c^2 - a^2
\][/tex]
5. Plug in the Known Values:
[tex]\[
b^2 = 8^2 - 2^2
\][/tex]
6. Calculate the Squares:
[tex]\[
8^2 = 64
\][/tex]
[tex]\[
2^2 = 4
\][/tex]
7. Subtract to Find \( b^2 \):
[tex]\[
b^2 = 64 - 4
\][/tex]
[tex]\[
b^2 = 60
\][/tex]
8. Take the Square Root to Find \( b \):
[tex]\[
b = \sqrt{60}
\][/tex]
9. Simplify the Square Root:
[tex]\[
\sqrt{60} \approx 7.746
\][/tex]
So, the distance from the door to where the truck should park is approximately [tex]\( 7.746 \)[/tex] feet.
