Math Problem Statement

Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown in the accompanying figure.

Solution

To find the dimensions of the rectangle of the largest area that can be inscribed in the given right triangle, we’ll use calculus and geometry. Let's break down the steps:

Step 1: Set Up the Geometry

The right triangle has legs of length 33 and 44 and a hypotenuse of length 55. We need to find the dimensions ww (width) and hh (height) of the inscribed rectangle that maximize the area.

Step 2: Establish a Relationship Between hh and ww

Since the rectangle is inscribed in the triangle, we can use similar triangles to relate hh and ww:

  1. The smaller triangle (with height hh and width 3w3 - w) is similar to the large right triangle (with height 44 and width 33).
  2. This similarity gives us the proportion: h4=3w3\frac{h}{4} = \frac{3 - w}{3}
  3. Solving for hh in terms of ww: h=43w3=443wh = 4 \cdot \frac{3 - w}{3} = 4 - \frac{4}{3}w

Step 3: Set Up the Area Function

The area AA of the rectangle is given by: A=whA = w \cdot h Substitute h=443wh = 4 - \frac{4}{3}w: A=w(443w)=4w43w2A = w \left(4 - \frac{4}{3}w\right) = 4w - \frac{4}{3}w^2

Step 4: Maximize the Area Function

To find the maximum area, we differentiate AA with respect to ww and set the derivative equal to zero:

  1. Differentiate: dAdw=483w\frac{dA}{dw} = 4 - \frac{8}{3}w
  2. Set the derivative to zero: 483w=04 - \frac{8}{3}w = 0
  3. Solve for ww: 83w=4    w=438=32\frac{8}{3}w = 4 \implies w = \frac{4 \cdot 3}{8} = \frac{3}{2}

Step 5: Find the Corresponding hh

Using h=443wh = 4 - \frac{4}{3}w: h=44332=42=2h = 4 - \frac{4}{3} \cdot \frac{3}{2} = 4 - 2 = 2

Conclusion

The dimensions of the rectangle that maximize the area are:

  • Width (ww) = 32\frac{3}{2}
  • Height (hh) = 22

Would you like further clarification on any part of this solution?


Here are some additional questions to explore this topic further:

  1. How would the solution change if the triangle had different dimensions?
  2. Could we solve this problem using another method, such as Lagrange multipliers?
  3. What is the area of the largest rectangle that can be inscribed in the given triangle?
  4. How does the concept of similar triangles help in maximizing the area?
  5. What if the rectangle was inscribed in an isosceles or scalene triangle?

Tip: When solving optimization problems, always check that you've used any symmetry or similar properties of shapes, as it often simplifies the work.

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Math Problem Analysis

Mathematical Concepts

Optimization
Similar Triangles
Differentiation

Formulas

Area of a rectangle A = w * h
Relationship from similar triangles: h / 4 = (3 - w) / 3

Theorems

Similarity of Triangles

Suitable Grade Level

Grades 11-12