srry. the question was I have a rectangular garden and I only have 24 meters of fence to go around it. so what is the largest area I can make the rectangular garden. thanks for your time and sorry for bothering! I just need help to figure this out.
A square, a specific type of rectangle, would give you the largest area.
No problem at all! I'm here to help you figure it out. Let's break it down step by step.
To find the largest area for a rectangular garden given a fixed perimeter, we need to determine the dimensions (length and width) that will maximize the area.
Let's label the length of the garden as 'L' and the width as 'W'. We're given that the total fence length is 24 meters, so we can set up an equation based on the perimeter:
Perimeter = 2L + 2W = 24.
To find the largest area, we can use the formula: Area = Length × Width.
Now, let's solve for one variable in terms of the other, which we can substitute back into the area formula to express it as a single-variable equation.
From the perimeter equation, we can isolate one variable by subtracting 2W from both sides:
2L = 24 - 2W.
Now, divide both sides by 2 to solve for L in terms of W:
L = (24 - 2W) / 2.
Substituting this value of L back into the area formula, we get:
Area = W × [(24 - 2W) / 2].
Simplifying this equation further, we have:
Area = W × (12 - W).
To maximize the area, we need to find the value of W that will yield the largest result. We can do this by finding the vertex of the quadratic equation, which corresponds to the maximum point of the graph.
The equation Area = W × (12 - W) represents a downward-opening parabola, and the value of W that maximizes the area is at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:
W = -b / 2a,
where a = -1 and b = 12. Plugging in these values, we get:
W = -12 / (2 * -1) = 6.
Since we cannot have negative values for the dimensions of a garden, we discard the negative value of W and consider only the positive value.
So, the width of the rectangular garden that maximizes the area is W = 6 meters.
Plugging this value back into the perimeter equation, we find:
2L + 2(6) = 24,
2L + 12 = 24,
2L = 12,
L = 6.
Therefore, the length of the rectangular garden is also L = 6 meters.
To summarize, the largest area you can make for the rectangular garden with a total fence length of 24 meters is when the length and width are both equal to 6 meters. The area in this case would be:
Area = Length × Width = 6 × 6 = 36 square meters.
So, the maximum area is 36 square meters.