A third ant walks around the perimeter of another rectangle sheet of paper. The dimensions of this sheet of paper are given in decimal numbers expressed to the hundredth place, with the tenths and hundredths digits being non-zero. If the ant travel between 18 and 19 centimeters, what are the dimensions of the sheet of paper? Give one possibility.
Let the length and width of the rectangle be $x$ and $y$ centimeters, respectively. The perimeter of the rectangle is $2x+2y$ centimeters, so the ant walks a distance of $2x+2y$ centimeters around the perimeter. We know that $18 < 2x + 2y < 19$.
Since $x$ and $y$ are given as decimal numbers expressed to the hundredth place, we know that they can be written in the form $a.b$, where $b$ is not equal to zero. Expressing $2x+2y$ in this form, we have $$2x+2y=2(a+b)+2(b+c)=2a+4b+2c.$$However, we want to express $2x+2y$ in the form $x.y$, where $y$ is not equal to zero. Taking the product of both sides by $0.1$ (which will shift the digits one place to the right), we obtain $$0.2x+0.2y=0.2a+0.4b+0.2c=a.b,$$from which it follows that $a+b$ is equal to $2a$, or $a=b$. Since neither $a$ nor $b$ can be zero, the only possibility is $a=b=1$. We plug this value back into the equation $0.2x+0.2y=0.2a+0.4b+0.2c$ and obtain $0.2x+0.2y=0.6$. Thus, $x+y=\boxed{3}.$
Let's assume that the length and width of the rectangle sheet of paper are given as L and W, respectively.
To find the dimensions, we need to determine the possible values for L and W given that the ant travels between 18 and 19 centimeters when walking around the perimeter of the rectangle.
The formula for calculating the perimeter of a rectangle is:
Perimeter = 2L + 2W
As the ant travels between 18 and 19 centimeters, we can set up the following inequality:
18 < 2L + 2W < 19
Since tenths and hundredths digits are non-zero, let's assume that L = 5.99 (5 and 99 hundredths) and substitute it into the inequality:
18 < 2(5.99) + 2W < 19
Simplifying the inequality:
18 < 11.98 + 2W < 19
Subtracting 11.98 from all parts of the inequality:
6.02 < 2W < 7.02
Dividing all parts of the inequality by 2:
3.01 < W < 3.51
Therefore, one possible set of dimensions for the rectangle sheet of paper is L = 5.99 units and W = 3.01 units.
To find the dimensions of the rectangle sheet of paper, we need to determine the range of possible lengths and widths, considering that the ant travels between 18 and 19 centimeters.
First, let's consider the length of the rectangle. We know that the ant walks around the perimeter of the sheet, so the distance it covers is equal to the sum of all four sides of the rectangle.
Let's denote the length of the rectangle as L and the width as W.
Considering that L and W are in decimal numbers, we can write the formula for the perimeter as:
Perimeter = 2L + 2W
Since the ant walks between 18 and 19 centimeters, we can write:
18 ≤ Perimeter ≤ 19
Substituting the formula for the perimeter, we get:
18 ≤ 2L + 2W ≤ 19
We need to find one possibility for the dimensions, so let's assume the width is minimum, with a value of 0.01 cm.
Substituting W = 0.01 into the inequality, we have:
18 ≤ 2L + 2(0.01) ≤ 19
Simplifying:
18 ≤ 2L + 0.02 ≤ 19
Subtracting 0.02 from all parts of the inequality:
17.98 ≤ 2L ≤ 18.98
Dividing all parts by 2:
8.99 ≤ L ≤ 9.49
Therefore, one possibility for the dimensions of the sheet of paper is a length (L) between 8.99 cm and 9.49 cm and a width (W) of 0.01 cm.