A rectangle has a perimeter of 16 meters and an area of 15 square meters. What is the longest of the side lengths, in meters, of the rectangle?

Options

A. 3
B. 5
C. 10
D. 15

Answer

To solve the problem, we need to find the dimensions of a rectangle given its perimeter and area.

**Step 1: Set up the equations for perimeter and area.**

The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2 \times (l + b) \]
where \( l \) is the length and \( b \) is the breadth. Given that the perimeter is 16 meters, we can write:
\[ 2 \times (l + b) = 16 \]
Dividing both sides by 2, we get:
\[ l + b = 8 \quad \text{(Equation 1)} \]

The area \( A \) of a rectangle is given by the formula:
\[ A = l \times b \]
Given that the area is 15 square meters, we can write:
\[ l \times b = 15 \quad \text{(Equation 2)} \]

**Step 2: Express one variable in terms of the other.**

From Equation 1, we can express \( b \) in terms of \( l \):
\[ b = 8 - l \]

**Step 3: Substitute into the area equation.**

Now, we substitute \( b \) in Equation 2:
\[ l \times (8 - l) = 15 \]
Expanding this gives:
\[ 8l - l^2 = 15 \]
Rearranging the equation, we get:
\[ l^2 - 8l + 15 = 0 \]

**Step 4: Solve the quadratic equation.**

To solve the quadratic equation \( l^2 - 8l + 15 = 0 \), we can factor it:
\[ (l - 3)(l - 5) = 0 \]
Setting each factor to zero gives us:
\[ l - 3 = 0 \quad \Rightarrow \quad l = 3 \]
\[ l - 5 = 0 \quad \Rightarrow \quad l = 5 \]

**Step 5: Find the corresponding breadth values.**

Using \( l = 3 \):
\[ b = 8 - 3 = 5 \]
Using \( l = 5 \):
\[ b = 8 - 5 = 3 \]

Thus, the dimensions of the rectangle are 3 meters and 5 meters.

**Step 6: Identify the longest side.**

The longest side length of the rectangle is:
\[ \text{Longest side} = 5 \text{ meters} \]

So, the final answer is:
\[ \text{The longest side length of the rectangle is } 5 \text{ meters.} \]

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