How many rectangles can be drawn with 38 cm as perimeter? Also, find the dimensions of the rectangle whose area will be maximum.
To find the number of rectangles that can be drawn with a given perimeter, we need to consider the possible combinations of dimensions.
A rectangle's perimeter is calculated by adding the lengths of all its sides. In this case, the perimeter is given as 38 cm. So we need to find all possible combinations of side lengths that add up to 38 cm.
Let's suppose one side of the rectangle is x cm, and the other side is y cm. Since opposite sides of a rectangle are equal, we can express the perimeter as:
2x + 2y = 38
We can simplify this equation further by dividing both sides by 2:
x + y = 19
Now we have a system of linear equations with two variables. To find the number of rectangles, we need to find all integer solutions to this equation.
Let's try various values of x and solve for y:
If x = 1, then y = 18
If x = 2, then y = 17
If x = 3, then y = 16
...
If x = 18, then y = 1
So, there are 18 rectangles that can be drawn with a perimeter of 38 cm.
Now let's find the dimensions of the rectangle that will give the maximum area. The area of a rectangle is given by length multiplied by width.
In our case, we want to maximize the area, which means we need to find the maximum possible product of two side lengths (x and y), given the constraint that their sum is 19.
To maximize the product xy, the two side lengths should be as close as possible. So, let's consider the dimensions where x and y are closest in value.
If x = 9 and y = 10, then the area = x * y = 9 * 10 = 90 cm².
Therefore, the rectangle with dimensions 9 cm and 10 cm has the maximum possible area when the perimeter is 38 cm.
To find the number of rectangles that can be drawn with a perimeter of 38 cm, we need to consider the possible combinations of side lengths.
Let's assume the length of the rectangle is L and the width is W.
According to the formula for the perimeter of a rectangle, which is 2(L + W), we can set up the equation: 2(L + W) = 38 cm.
Simplifying this equation, we get L + W = 19 cm.
Now, let's list down the possible combinations of side lengths that satisfy this equation:
L = 1 cm and W = 18 cm
L = 2 cm and W = 17 cm
L = 3 cm and W = 16 cm
L = 4 cm and W = 15 cm
L = 5 cm and W = 14 cm
L = 6 cm and W = 13 cm
L = 7 cm and W = 12 cm
L = 8 cm and W = 11 cm
L = 9 cm and W = 10 cm
So, there are 9 possible rectangles that can be drawn with a perimeter of 38 cm.
To find the dimensions of the rectangle with the maximum area, we need to find the combination of side lengths that gives the largest possible product, as area is given by the formula A = L * W.
To do this, let's calculate the product (L * W) for each of the above combinations:
1 cm * 18 cm = 18 cm²
2 cm * 17 cm = 34 cm²
3 cm * 16 cm = 48 cm²
4 cm * 15 cm = 60 cm²
5 cm * 14 cm = 70 cm²
6 cm * 13 cm = 78 cm²
7 cm * 12 cm = 84 cm²
8 cm * 11 cm = 88 cm²
9 cm * 10 cm = 90 cm²
From the calculations, we can see that the rectangle with dimensions 9 cm by 10 cm will have the maximum area of 90 square cm.