A string is 12in. long which makes a rectangle area of 5sq.inch can you make a shape that has a greater area with the string, if so what's the shape and area.
The shape giving you the largest area is a circle
so 2πr = 12
r = 6/π
area = πr^2 = π(36/π^2) = 36/π
or appr 11.46 , more than double the area of the rectangle
If you had to stay with a rectangle, then with a perimeter of 12, we could have used a square with sides 3 units.
then the area would have been 9 inches^2
To determine if there is a shape with a greater area that can be formed using the given string length, we need to consider the properties of different shapes.
First, let's calculate the perimeter of the given rectangle. The perimeter is the sum of all the sides. Since the rectangle has two pairs of equal sides, we can calculate the perimeter as follows:
Perimeter of rectangle = 2 * (length + width)
Given that the length of the string is 12 inches, we can rewrite the equation as:
12 = 2 * (length + width)
Simplifying the equation, we get:
6 = length + width
Now, let's find the area of the rectangle. The area of a rectangle is calculated by multiplying the length and width:
Area of rectangle = length * width
Given that the area of the rectangle is 5 square inches, we have:
5 = length * width
Now we have two equations describing the relationship between the length and width. We can solve these equations simultaneously to find the values of length and width.
From the equation 6 = length + width, we can rearrange it to:
length = 6 - width
Substituting this value of length into the equation 5 = length * width, we get:
5 = (6 - width) * width
Expanding the equation, we have:
5 = 6w - w^2
Rearranging the equation and converting it to a quadratic equation, we get:
w^2 - 6w + 5 = 0
Now we can solve this quadratic equation to find the possible values of width (w). By factoring or using the quadratic formula, we find that the possible values for width are:
w = 5
w = 1
Since the width of a shape cannot be greater than its perimeter divided by 4 (in this case, 3), we choose w = 1 as the width for further calculations.
Now, substituting the value of width into the equation length = 6 - width, we get:
length = 6 - 1 = 5
Therefore, using the given string length of 12 inches, a shape with a greater area can be formed using a rectangle with a length of 5 inches and a width of 1 inch.
To calculate the area of this rectangle, we multiply the length and width:
Area = length * width = 5 * 1 = 5 square inches.
Hence, the shape with the greater area that can be formed using the given string is a rectangle with a length of 5 inches and a width of 1 inch, resulting in an area of 5 square inches.