44 16 22 8. Scott is using a 12-foot ramp to help load furniture into the back of a moving truck. If the back of the truck is 3.5 feet from the ground, what is the horizontal distance from where the ramp reaches the ground to the truck? Cha Wilson (All Things Algebra, LLC). 2014-2018
The Correct Answer and Explanation is:
To solve this problem, we can model it as a right triangle:
- The ramp is the hypotenuse of the right triangle and measures 12 feet.
- The height from the ground to the back of the truck is one leg of the triangle and is 3.5 feet.
- The horizontal distance from the bottom of the ramp to the truck is the other leg of the triangle, which we need to find.
We can use the Pythagorean Theorem to solve this: a2+b2=c2a^2 + b^2 = c^2
Where:
- aa = horizontal distance (what we are solving for),
- bb = vertical height (3.5 feet),
- cc = hypotenuse (12 feet).
Plug in the known values: a2+3.52=122a^2 + 3.5^2 = 12^2 a2+12.25=144a^2 + 12.25 = 144
Subtract 12.25 from both sides: a2=144−12.25=131.75a^2 = 144 – 12.25 = 131.75
Now take the square root: a=131.75≈11.48a = \sqrt{131.75} \approx 11.48
✅ Correct Answer: Approximately 11.5 feet
Explanation (300+ Words):
This scenario involves right triangle geometry, which often comes up in real-life applications like ramps, stairs, and ladders. In this case, Scott is using a ramp to load furniture into a truck, forming a triangle with the ground (horizontal side), the ramp (slanted hypotenuse), and the height of the truck’s back (vertical side).
To solve for the unknown horizontal distance (the leg of the triangle along the ground), we use the Pythagorean Theorem, a fundamental concept in geometry. This theorem states that in a right 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.
Given:
- Hypotenuse = 12 feet
- Height = 3.5 feet
We substitute into the formula: a2=122−3.52=144−12.25=131.75a^2 = 12^2 – 3.5^2 = 144 – 12.25 = 131.75
Then we take the square root to find the actual distance: a≈131.75≈11.48 feeta \approx \sqrt{131.75} \approx 11.48 \text{ feet}
This tells us that the base of the ramp extends approximately 11.5 feet from the truck.
Understanding this problem helps in fields like construction, architecture, and physical therapy, where safe slope and distance calculations are essential. It also shows how math is not just theoretical but practical in everyday tasks like moving furniture safely into a truck.