Math Problem Statement

A fence 8 feet tall runs parallel to a tall building at a distance of 4 ft from the building as shown in the diagram. LADDER4 ft8 ftO[Graphs generated by this script: initPicture(-10,110,-10);line([0,-1],[100,-1]);stroke="red";fill="red";rect([80,0],[100,100]);stroke="blue";strokewidth=2;line([60,0],[60,53]);stroke="lightgreen";strokewidth=5;line([0,0],[80,70]);stroke="black";strokewidth=1;line([61,6],[80,6]);line([63,8],[61,6]);line([63,4],[61,6]);line([77,8],[79,6]);line([77,4],[79,6]);line([1,6],[79,74]);line([2,10],[1,6]);line([4,6],[1,6]);line([76,74],[79,74]);line([78,70],[79,74]);line([58,0],[58,50]);line([58,0],[60,2]);line([58,0],[56,2]);line([58,50],[60,48]);line([58,50],[56,48]);line([8,3],[12,3]);text([35,50],"LADDER");text([70,11],"4 ft");text([50,22],"8 ft");text([10,3],"O")]

We wish to find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.

[A] First, find a formula for the length of the ladder in terms of . (Hint: split the ladder into 2 parts.) Type theta for .

Solution

To find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building, let’s analyze the geometry involved in the setup. We will split the ladder into two parts: the part extending horizontally from the wall to the fence, and the part reaching over the fence to the ground.

Let:

  • θ\theta be the angle between the ladder and the ground.
  • LL be the length of the ladder we need to find.

We can split LL into two segments:

  1. The horizontal segment from the wall to the top of the fence, which we’ll call xx.
  2. The hypotenuse from the top of the fence down to the ground, which we’ll call L2L_2.

The geometry of the situation gives us two main relationships:

  1. For the segment reaching from the ground to the top of the fence (diagonal segment): The ladder forms a right triangle with the ground and the height of the fence. Using trigonometry: tan⁡(θ)=84\tan(\theta) = \frac{8}{4} where 8 feet is the height of the fence, and 4 feet is the horizontal distance from the building to the fence.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangle Geometry
Optimization

Formulas

tan(θ) = opposite/adjacent
L = L1 + L2, where L1 is the horizontal distance and L2 is the length over the fence

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 11-12

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