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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 \(3\) and \(4\) and a hypotenuse of length \(5\). We need to find the dimensions \(w\) (width) and \(h\) (height) of the inscribed rectangle that maximize the area.

### Step 2: Establish a Relationship Between \(h\) and \(w\)
Since the rectangle is inscribed in the triangle, we can use similar triangles to relate \(h\) and \(w\):
1. The smaller triangle (with height \(h\) and width \(3 - w\)) is similar to the large right triangle (with height \(4\) and width \(3\)).
2. This similarity gives us the proportion:
   \[
   \frac{h}{4} = \frac{3 - w}{3}
   \]
3. Solving for \(h\) in terms of \(w\):
   \[
   h = 4 \cdot \frac{3 - w}{3} = 4 - \frac{4}{3}w
   \]

### Step 3: Set Up the Area Function
The area \(A\) of the rectangle is given by:
\[
A = w \cdot h
\]
Substitute \(h = 4 - \frac{4}{3}w\):
\[
A = w \left(4 - \frac{4}{3}w\right) = 4w - \frac{4}{3}w^2
\]

### Step 4: Maximize the Area Function
To find the maximum area, we differentiate \(A\) with respect to \(w\) and set the derivative equal to zero:
1. Differentiate:
   \[
   \frac{dA}{dw} = 4 - \frac{8}{3}w
   \]
2. Set the derivative to zero:
   \[
   4 - \frac{8}{3}w = 0
   \]
3. Solve for \(w\):
   \[
   \frac{8}{3}w = 4 \implies w = \frac{4 \cdot 3}{8} = \frac{3}{2}
   \]

### Step 5: Find the Corresponding \(h\)
Using \(h = 4 - \frac{4}{3}w\):
\[
h = 4 - \frac{4}{3} \cdot \frac{3}{2} = 4 - 2 = 2
\]

### Conclusion
The dimensions of the rectangle that maximize the area are:
- Width (\(w\)) = \( \frac{3}{2} \)
- Height (\(h\)) = \(2\)

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

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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.

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

Explore how to determine the dimensions of the largest rectangle that can be inscribed within a right triangle with legs of 3 and 4 units. This problem involves optimization, differentiation, and similar triangles, providing a detailed step-by-step solution for students in Grades 11-12.

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