A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? (Round your answer to two decimal places.)

To find the shortest ladder length that will reach from the ground over the fence to the wall of the building, we can solve this problem using calculus and geometric relationships.

Let's break down the problem:

1. **Given values**:
   - The fence is 8 feet tall.
   - The fence is 4 feet away from the building.

2. **Objective**: Find the shortest length of a ladder that can reach over the fence and touch the wall of the building.

Let’s define some variables and use calculus for optimization:

### Step 1: Set up the Geometry
Let:
- \( x \) be the horizontal distance from the ladder's bottom end (on the ground) to the fence.
- \( L \) be the total length of the ladder.

From the setup, the ladder reaches over the fence to touch the building. The geometry forms a right triangle, with \( L \) as the hypotenuse.

### Step 2: Express Ladder Length \( L \) in Terms of \( x \)
For the ladder to just clear the fence and reach the building wall:
1. The ladder’s height at the distance \( x + 4 \) (the total distance from the bottom of the ladder to the building) must be enough to pass over the 8-foot-high fence.

2. By similar triangles, we can express the height of the ladder \( y \) at the point where it clears the fence.

Using the proportion between the height \( y \) and base \( x \), and the total height and base of the ladder triangle, we get:

\[
y = \frac{8}{x}

Discover how to calculate the shortest ladder length to clear an 8-foot fence located 4 feet from a building using calculus and geometry. This optimization problem involves the Pythagorean theorem and is ideal for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation