The perimeter of a rectangle is 540ft. Let x represent the width of the rectangle. Write a quadratic function for the area of rectangle in terms of its width. Find the vertex of the graph of the quadratic function and interpret its meaning in the context of the problem.
the length is 270-x
So, the area is length*width, or
f(x) = x(270-x)
As with all parabolas, the vertex is midway between the roots, so in this case, it's at x=135.
Since the graph opens downward, the vertex represents the maximum possible area.
Oh, quadratic functions and rectangles, what a perfect combo for a clown like me! 🎪
To find the quadratic function for the area of a rectangle in terms of its width, we must first understand that the area of a rectangle is given by A = length × width. In this case, we are only concerned with the width, so let's express the length in terms of x as well.
Since the perimeter is the sum of all sides, we have:
540 ft = 2(length) + 2(width)
540 ft = 2(length) + 2x
Simplifying a bit, we get:
270 ft = length + x
length = 270 ft - x
Now, let's substitute the expressions for length and width into the area formula:
A = (270 ft - x) * x
A = 270x - x^2
Ta-da! We have our quadratic function for the area! 🎉
To find the vertex of this quadratic function, we remember that the vertex occurs at the maximum or minimum point of the graph. In this case, since we're talking about the area of a rectangle, it will be a maximum point.
The vertex of a quadratic function can be found using the formula:
x = -b / (2a)
In our case, a = -1 and b = 270.
x = -270 / (2 * -1)
x = 270 / 2
x = 135
So, the width of the rectangle at the vertex is 135 ft.
Now, let me interpret its meaning in this delightful context! 🤡
The vertex represents the maximum area that can be achieved with a rectangle of given perimeter. In this case, it means that with a perimeter of 540 ft, the rectangle with a width of 135 ft will have the largest possible area. Keep in mind that this is just a clown's interpretation, but it's always good to maximize your rectangle's area when you can! 🎈
Let's first find the length of the rectangle in terms of its width x.
We know that the perimeter of a rectangle is given by the formula:
Perimeter = 2 * (Length + Width)
Since the perimeter is given as 540ft, we can write the equation:
540 = 2 * (Length + x)
Dividing both sides by 2, we get:
270 = Length + x
Rearranging the equation, we find:
Length = 270 - x
Now let's write the quadratic function for the area of the rectangle in terms of its width x.
The area of a rectangle is given by the formula:
Area = Length * Width
Substituting the value we found for the length, we get:
Area = (270 - x) * x
Expanding the equation, we have:
Area = 270x - x^2
This quadratic function represents the area of the rectangle in terms of its width.
To find the vertex of the graph of this quadratic function, we can use the formula:
x = -b / (2a)
In this case, a = -1 and b = 270.
Substituting these values into the formula, we get:
x = -270 / (2 * -1)
x = -270 / -2
x = 135
The x-coordinate of the vertex is 135.
To interpret the meaning of the vertex in the context of the problem, the width represented by x is 135ft. This means that when the width of the rectangle is 135ft, the area of the rectangle will be at its maximum.
To write a quadratic function for the area of a rectangle in terms of its width, we need to first understand the formula for the perimeter and area of a rectangle.
The formula for the perimeter of a rectangle is:
Perimeter = 2 * (Length + Width)
Since we know that the perimeter of the rectangle is 540ft, we can set up the equation:
540 = 2 * (Length + x)
Now, let's simplify the equation:
270 = Length + x
Next, we'll use the formula for the area of a rectangle:
Area = Length * Width
We can substitute the value of Length from the previous equation into the area formula:
Area = (270 - x) * x
Now we have a quadratic function for the area of the rectangle in terms of its width: Area = -x^2 + 270x
To find the vertex of the quadratic function, we can use the formula:
Vertex x-coordinate = -b / 2a
For our quadratic function, the coefficient of x^2 is -1 (a=-1) and the coefficient of x is 270 (b=270). Substituting these values into the formula:
Vertex x-coordinate = -270 / (2 * -1) = 135
To find the y-coordinate of the vertex, we substitute the x-coordinate back into the quadratic function:
Area = -(135)^2 + 270 * 135
Area = -18225 + 36450
Area = 18225
Therefore, the vertex of the graph of the quadratic function is (135, 18225).
Interpreting the meaning of the vertex in the context of the problem, the x-coordinate 135 represents the width of the rectangle that results in the maximum area. The y-coordinate 18225 represents the maximum area of the rectangle when the width is 135 ft.