We are given that the area of a rectangle is 2900 square meters, and one side is 8 meters longer than the other. We need to find the dimensions of the rectangle.

Let:

• $x$ be the length of the shorter side of the rectangle (in meters).
• The longer side would then be $x + 8$ meters (since it is 8 meters longer than the shorter side).

The formula for the area of a rectangle is:

$\text{Area} = \text{length} \times \text{width}$

Substitute the known values into the equation:

$2900 = x(x + 8)$

Now, expand and solve the quadratic equation:

$2900 = x^2 + 8x$

Rearrange the terms:

$x^2 + 8x - 2900 = 0$

To solve this quadratic equation, we will use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $x^2 + 8x - 2900 = 0$, the coefficients are:

• $a = 1$
• $b = 8$
• $c = -2900$

Substitute these values into the quadratic formula:

$x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-2900)}}{2(1)}$

$x = \frac{-8 \pm \sqrt{64 + 11600}}{2}$

$x = \frac{-8 \pm \sqrt{11664}}{2}$

$x = \frac{-8 \pm 108}{2}$

Now, we have two possible solutions:

1. $x = \frac{-8 + 108}{2} = \frac{100}{2} = 50$
2. $x = \frac{-8 - 108}{2} = \frac{-116}{2} = -58$

Since the length of a side cannot be negative, we discard $x = -58$ and accept $x = 50$.

Thus, the shorter side is 50 meters, and the longer side is:

$50 + 8 = 58 \text{ meters}$

Final Answer:

• The dimensions of the rectangle are 50 meters by 58 meters.

Would you like a more detailed explanation or have any questions? Here are some related questions:

1. How would the solution change if the area were different, say 3600 square meters?
2. How can we find the dimensions of a rectangle if only the perimeter is given instead of the area?
3. What happens to the dimensions of the rectangle if the difference between the sides is changed to 10 meters?
4. Can we solve this kind of problem using a system of equations with different relationships between the sides?
5. How do we graph a quadratic equation like the one we solved to visualize the solutions?

Tip: Always check if both solutions from a quadratic equation are physically meaningful in the context of the problem (in this case, side lengths cannot be negative).