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Answer :
Sure, let's break down the solution step by step for each part of the question:
(3) Finding the Length of the Garden:
1. We know the garden is rectangular with a width of 90 feet and a perimeter of 500 feet.
2. The formula for the perimeter of a rectangle is [tex]\( P = 2L + 2W \)[/tex].
3. Plugging the known values into this formula:
[tex]\( 500 = 2 \times \text{length} + 2 \times 90 \)[/tex].
4. Simplifying, we have [tex]\( 500 = 2 \times \text{length} + 180 \)[/tex].
5. Subtract 180 from both sides:
[tex]\( 320 = 2 \times \text{length} \)[/tex].
6. Divide by 2 to find the length:
[tex]\( \text{length} = 160 \)[/tex] feet.
(4) Determining if Amy Has Enough Ribbon:
1. Amy has a ribbon that is 36 inches long. We need to check if this ribbon is enough to wrap around the frames with the given shapes and sizes.
- Square with side lengths of 9 inches:
- Perimeter = [tex]\( 4 \times 9 = 36 \)[/tex] inches.
- Amy can wrap this because 36 inches is just enough.
- Rectangle with dimensions 18 inches by 10 inches:
- Perimeter = [tex]\( 2 \times (18 + 10) = 56 \)[/tex] inches.
- Amy cannot wrap this because 56 inches is more than 36 inches.
- Rectangle with dimensions 12 inches by 24 inches:
- Perimeter = [tex]\( 2 \times (12 + 24) = 72 \)[/tex] inches.
- Amy cannot wrap this because 72 inches is more than 36 inches.
- Square with side lengths of 6 inches:
- Perimeter = [tex]\( 4 \times 6 = 24 \)[/tex] inches.
- Amy can wrap this because 24 inches is less than 36 inches.
(5) Calculating Side Lengths for a Square and Rectangular Perimeter of 200 cm:
1. For the Square:
- The perimeter of the square is 200 centimeters.
- The formula for the perimeter of a square is [tex]\( P = 4 \times \text{side length} \)[/tex].
- Solving for side length gives [tex]\( \text{side length} = 200 \div 4 = 50 \)[/tex] cm.
2. For the Rectangle:
- The perimeter of the rectangle is also 200 centimeters.
- Let's assume possible dimensions for the rectangle.
- To find two potential side lengths, let's assume one side (width) can be half the length from earlier:
- If width is 25 cm, (as deduced alongside provided calculations),
- Then the length can be found using the formula:
- [tex]\( 200 = 2 \times \text{length} + 2 \times 25 \)[/tex].
- Simplifying gives [tex]\( 200 = 2 \times \text{length} + 50 \)[/tex].
- Subtract 50 from both sides:
[tex]\( 150 = 2 \times \text{length} \)[/tex].
- Divide by 2 gives [tex]\( \text{length} = 75 \)[/tex] cm.
In summary, the length of the garden is 160 feet, Amy can wrap the ribbon around the 9-inch square and the 6-inch square frames, and the side of the square is 50 cm, while the rectangle has a length of 75 cm and a width of 25 cm.
(3) Finding the Length of the Garden:
1. We know the garden is rectangular with a width of 90 feet and a perimeter of 500 feet.
2. The formula for the perimeter of a rectangle is [tex]\( P = 2L + 2W \)[/tex].
3. Plugging the known values into this formula:
[tex]\( 500 = 2 \times \text{length} + 2 \times 90 \)[/tex].
4. Simplifying, we have [tex]\( 500 = 2 \times \text{length} + 180 \)[/tex].
5. Subtract 180 from both sides:
[tex]\( 320 = 2 \times \text{length} \)[/tex].
6. Divide by 2 to find the length:
[tex]\( \text{length} = 160 \)[/tex] feet.
(4) Determining if Amy Has Enough Ribbon:
1. Amy has a ribbon that is 36 inches long. We need to check if this ribbon is enough to wrap around the frames with the given shapes and sizes.
- Square with side lengths of 9 inches:
- Perimeter = [tex]\( 4 \times 9 = 36 \)[/tex] inches.
- Amy can wrap this because 36 inches is just enough.
- Rectangle with dimensions 18 inches by 10 inches:
- Perimeter = [tex]\( 2 \times (18 + 10) = 56 \)[/tex] inches.
- Amy cannot wrap this because 56 inches is more than 36 inches.
- Rectangle with dimensions 12 inches by 24 inches:
- Perimeter = [tex]\( 2 \times (12 + 24) = 72 \)[/tex] inches.
- Amy cannot wrap this because 72 inches is more than 36 inches.
- Square with side lengths of 6 inches:
- Perimeter = [tex]\( 4 \times 6 = 24 \)[/tex] inches.
- Amy can wrap this because 24 inches is less than 36 inches.
(5) Calculating Side Lengths for a Square and Rectangular Perimeter of 200 cm:
1. For the Square:
- The perimeter of the square is 200 centimeters.
- The formula for the perimeter of a square is [tex]\( P = 4 \times \text{side length} \)[/tex].
- Solving for side length gives [tex]\( \text{side length} = 200 \div 4 = 50 \)[/tex] cm.
2. For the Rectangle:
- The perimeter of the rectangle is also 200 centimeters.
- Let's assume possible dimensions for the rectangle.
- To find two potential side lengths, let's assume one side (width) can be half the length from earlier:
- If width is 25 cm, (as deduced alongside provided calculations),
- Then the length can be found using the formula:
- [tex]\( 200 = 2 \times \text{length} + 2 \times 25 \)[/tex].
- Simplifying gives [tex]\( 200 = 2 \times \text{length} + 50 \)[/tex].
- Subtract 50 from both sides:
[tex]\( 150 = 2 \times \text{length} \)[/tex].
- Divide by 2 gives [tex]\( \text{length} = 75 \)[/tex] cm.
In summary, the length of the garden is 160 feet, Amy can wrap the ribbon around the 9-inch square and the 6-inch square frames, and the side of the square is 50 cm, while the rectangle has a length of 75 cm and a width of 25 cm.
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