2.10 Q-4

Question Statement

The perimeter of a triangle is 16 cm, and one side has a length of 6 cm. What is the length of the other side that maximizes the area of the triangle?

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Background and Explanation

This problem involves finding the side length that maximizes the area of a triangle when the perimeter is fixed. To solve it, we will use Heron’s formula for the area of a triangle and differentiate the area function to find the value of $x$ that gives the maximum area. The perimeter constraint gives us a relationship between the sides of the triangle, allowing us to set up the area function in terms of a single variable.

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Solution

1. Define the variables:

   Let the unknown side of the triangle be $x$ (in cm). Then, the other unknown side will be $16 - 6 - x = 10 - x$, since the perimeter is fixed at 16 cm.

2. Set up the area function:

   Using Heron’s formula, the area $A$ of the triangle is given by:

$$A = \sqrt{S(S - 6)(S - x)(S - (10 - x))}$$

where $S$ is the semi-perimeter of the triangle. Since the perimeter is 16 cm, the semi-perimeter is:

$$S = \frac{16}{2} = 8$$

Thus, the area function becomes:

$$y = 8(8 - 6)(8 - x)(8 - 10 + x)$$

Simplifying:

$$y = 8 \times 2 \times (8 - x) \times (x - 2)$$

This is the function we need to maximize.

3. Differentiate the area function:

   Now, we differentiate $y$ with respect to $x$ to find the critical points:

$$\frac{dy}{dx} = 16\left[(-1)(x - 2) + (8 - x)(1)\right]$$

Simplifying:

$$\frac{dy}{dx} = 16[2 - x + (8 - x)]$$ $$\frac{dy}{dx} = 16(10 - 2x)$$

4. Set the derivative equal to zero:

   To find the critical points, set $\frac{dy}{dx} = 0$:

$$16(10 - 2x) = 0$$

Solving for $x$:

$$10 - 2x = 0 \quad \Rightarrow \quad x = 5$$

5. Check the second derivative:

   To confirm that this value of $x$ gives a maximum, we compute the second derivative:

$$\frac{d^2y}{dx^2} = \frac{d}{dx}[16(10 - 2x)] = -32$$

Since $\frac{d^2y}{dx^2} = -32$, which is negative, we confirm that $x = 5$ gives a maximum.

6. Find the length of the other side:

   The length of the other side of the triangle is $10 - x = 10 - 5 = 5$ cm.

Thus, the length of the other side that maximizes the area of the triangle is 5 cm.

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Key Formulas or Methods Used

• Semi-perimeter: $S = \frac{\text{Perimeter}}{2}$
• Area of a triangle (Heron’s formula): $A = \sqrt{S(S - a)(S - b)(S - c)}$
• First derivative test: To find critical points and maximize the area.
• Second derivative test: To confirm that the critical point is a maximum.

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Summary of Steps

1. Define the unknown side as $x$, and express the other side as $10 - x$.
2. Use Heron’s formula to set up the area function in terms of $x$.
3. Differentiate the area function to find $\frac{dy}{dx}$.
4. Set the derivative equal to zero to find $x = 5$.
5. Compute the second derivative to confirm the maximum.
6. The length of the other side is $5$ cm when the area is maximized.