Complete the activity. An ant stands at one vertex of a rectangular sheet of paper. The ant then proceeds to walk all the way around the perimeter of the paper. The dimensions of the sheet of paper are 358 inches by 278 inches. How many inches does the ant travel? Show your work. Suppose that each dimension of the sheet of paper described in question 1 is increased by one inch. How many inches will the ant travel in this case? Explain your thinking. A second ant walks all the way around a triangle drawn in the dirt. Two of the dimensions of the triangle are 5110 centimeters and 715 centimeters. The total distance traveled by the ant is 20 centimeters. What is the length of the third side of the triangle? A third ant walks around the perimeter of another rectangular 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 travels between 18 and 19 centimeters, what are the dimensions of the sheet of paper? Give one possibility.
To find the distance the ant travels around the rectangular sheet of paper, we can use the formula for the perimeter of a rectangle: P = 2(length + width).
Given the dimensions of the paper, length = 358 inches and width = 278 inches.
Plugging these values into the formula, we get:
P = 2(358 + 278)
P = 2(636)
P = 1272 inches
Therefore, the ant travels 1272 inches around the rectangular sheet of paper.
If each dimension of the sheet of paper is increased by one inch, the new dimensions would be length = 359 inches and width = 279 inches.
Using the same formula, we can calculate the new distance the ant travels around the paper:
P = 2(359 + 279)
P = 2(638)
P = 1276 inches
Therefore, the ant would travel 1276 inches in this case.
Let the third side of the triangle be x centimeters.
The perimeter of the triangle is the sum of all three sides, which is given as 20 centimeters.
So, we have the equation:
5110 + 715 + x = 20
5825 + x = 20
x = 20 - 5825
x = -5805 centimeters
Since negative length doesn't make sense, there seems to be an error in the information provided. It's not possible for a triangle with side lengths 5110 and 715 centimeters to have a combined perimeter of 20 centimeters.
Let the dimensions of the rectangular sheet of paper be length = a centimeters and width = b centimeters.
From the problem, we know that the perimeter is between 18 and 19 centimeters, so we can set up the inequality:
2(a + b) > 18 and 2(a + b) < 19
Dividing both sides of the inequalities by 2, we get:
a + b > 9 and a + b < 9.5
Since the dimensions must be expressed in decimal numbers to the hundredth place, and the tenths and hundredths digits are non-zero, we can have a = 4.50 and b = 4.50.
This would give us:
2(4.50 + 4.50) = 2(9) = 18 centimeters
Therefore, one possibility for the dimensions of the sheet of paper is 4.50 centimeters by 4.50 centimeters.
Sure! Let's solve each of these problems step-by-step:
1. Given that the dimensions of the rectangular sheet of paper are 358 inches by 278 inches, we need to find the total distance the ant travels around the perimeter.
The perimeter of a rectangle is found using the formula: Perimeter = 2 * (Length + Width).
So, for this rectangle, the perimeter = 2 * (358 + 278) = 2 * 636 = 1272 inches.
Therefore, the ant travels 1272 inches.
2. If each dimension of the sheet of paper described in question 1 is increased by one inch, the perimeter of the new rectangle will be calculated as follows:
New perimeter = 2 * (Length + 1 + Width + 1) = 2 * (358 + 1 + 278 + 1) = 2 * 638 = 1276 inches.
Hence, the ant will travel 1276 inches.
3. Let's denote the length of the third side of the triangle as x. The total distance traveled by the ant is given as 20 centimeters.
The perimeter of a triangle is found by adding all three sides. In this case, the perimeter is given as 20 centimeters.
So, we can set up the equation as: 5110 + 715 + x = 20.
Combining the given dimensions, we get: 5825 + x = 20.
Solving for x, we find x = 20 - 5825 = -5805 (negative result).
Since length cannot be negative, it seems there might be an error in the given information.
4. The dimensions of the rectangular sheet of paper are given between 18 and 19 centimeters.
Let's assume the length of the paper is 18.05 cm and the width is 0.95 cm.
The perimeter of the rectangle is calculated using the formula: Perimeter = 2 * (Length + Width).
So, the perimeter = 2 * (18.05 + 0.95) = 2 * 19 = 38 centimeters.
Therefore, the dimensions of one possible sheet of paper could be 18.05 cm by 0.95 cm, and the ant will travel 38 centimeters.
To find the distance traveled by the ant in the first scenario, we can use the formula for the perimeter of a rectangle: P = 2(length + width). The given dimensions are 358 inches by 278 inches. Plugging the values into the formula:
P = 2(358 + 278)
P = 2(636)
P = 1272 inches
So, the ant travels 1272 inches.
To find the distance traveled by the ant when each dimension is increased by one inch, we need to add 1 to both the length and width. The new dimensions would be 359 inches by 279 inches. Using the same formula as above:
P = 2(359 + 279)
P = 2(638)
P = 1276 inches
So, the ant would travel 1276 inches in this case.
Now, let's move on to the second scenario. The ant walks all the way around a triangle with two known dimensions of 5110 centimeters and 715 centimeters. The total distance traveled by the ant is 20 centimeters. We need to find the length of the third side of the triangle.
To find the perimeter of the triangle, we add up all three sides. We have two known sides (5110 cm and 715 cm), and the third side is unknown. The perimeter is given as 20 cm.
P = 5110 + 715 + x, where x represents the length of the third side.
Since the perimeter is 20 cm, we can set up the equation:
20 = 5110 + 715 + x
We simplify:
20 = 5825 + x
Next, isolate x:
x = 20 - 5825
x = -5805
However, it doesn't make sense for the length of a side to be negative. So, it seems there may be an error in the question or the given values. Please double-check the information provided or clarify any confusion.
Lastly, in the third scenario, the ant walks around the perimeter of another rectangular sheet of paper. The dimensions of this sheet are given as decimal numbers expressed to the hundredth place, with the tenths and hundredths digits being non-zero. The ant travels between 18 and 19 centimeters. We need to find the possible dimensions of the sheet of paper.
Let's consider the length and width as variables, l and w. The perimeter formula for a rectangle can be used: P = 2(l + w).
We can set up two inequalities based on the given information:
18 ≤ 2(l + w) ≤ 19
Divide both sides by 2:
9 ≤ l + w ≤ 9.5
Since the dimensions are given to the hundredth place, with non-zero tenths and hundredths digits, one possible solution could be l = 4.50 cm and w = 4.50 cm. This way, the perimeter would be:
P = 2(4.50 + 4.50) = 2(9) = 18 cm
So, one possibility is a rectangular sheet of paper with dimensions 4.50 cm by 4.50 cm.
Please note that there could be other valid possibilities depending on the specific values within the given range.