give the dimensions of rectangles with perimeters of 70ft and length to width ratios of 3to4 4to5 and 1to1
To find the dimensions of rectangles with different ratios and perimeters, we need to use the formulas for perimeter and the ratio of length to width.
1. Length to Width ratio of 3:4:
Let the length be 3x and the width be 4x.
The perimeter of a rectangle is given by: P = 2(l + w)
Substituting the values into the formula, we have:
70 = 2(3x + 4x)
70 = 2(7x)
35 = 7x
x = 35/7
x = 5
Therefore, the length is 3x = 3(5) = 15ft, and the width is 4x = 4(5) = 20ft.
2. Length to Width ratio of 4:5:
Let the length be 4y and the width be 5y.
Using the same formula: P = 2(l + w)
70 = 2(4y + 5y)
70 = 2(9y)
35 = 9y
y = 35/9
Therefore, the length is 4y = 4(35/9) ≈ 15.56ft, and the width is 5y = 5(35/9) ≈ 19.44ft.
3. Length to Width ratio of 1:1:
Let both length and width be x.
Again, using the perimeter formula: P = 2(l + w)
70 = 2(x + x)
70 = 2(2x)
35 = 2x
x = 35/2
Therefore, both the length and width are approximately 17.5ft.
To summarize:
- For a length to width ratio of 3:4, the dimensions are approximately 15ft by 20ft.
- For a length to width ratio of 4:5, the dimensions are approximately 15.56ft by 19.44ft.
- For a length to width ratio of 1:1, the dimensions are approximately 17.5ft by 17.5ft.
To find the dimensions of rectangles with different length to width ratios and a given perimeter, we can use the following steps:
Step 1: Calculate the perimeter of the rectangle.
Perimeter of a rectangle = 2 × (length + width)
Step 2: Determine the length and width ratios.
For the given ratios:
- 3 to 4: length = 3x, width = 4x
- 4 to 5: length = 4y, width = 5y
- 1 to 1: length = z, width = z
Step 3: Substitute the length and width values into the perimeter formula, and solve for x, y, and z.
Let's calculate the dimensions for each case:
Case 1: Length to width ratio of 3 to 4.
For this case, the length is 3x and the width is 4x.
Perimeter = 70 ft
2 × (length + width) = 70
2 × (3x + 4x) = 70
2 × 7x = 70
14x = 70
x = 70 / 14
x = 5
Length = 3x = 3 × 5 = 15 ft
Width = 4x = 4 × 5 = 20 ft
Therefore, for a perimeter of 70 ft and a length to width ratio of 3 to 4, the dimensions of the rectangle are 15 ft by 20 ft.
Case 2: Length to width ratio of 4 to 5.
For this case, the length is 4y and the width is 5y.
Perimeter = 70 ft
2 × (length + width) = 70
2 × (4y + 5y) = 70
2 × 9y = 70
18y = 70
y = 70 / 18
Using approximations:
y ≈ 3.89
Length = 4y ≈ 4 × 3.89 ≈ 15.56 ft
Width = 5y ≈ 5 × 3.89 ≈ 19.45 ft
Therefore, for a perimeter of 70 ft and a length to width ratio of 4 to 5, the dimensions of the rectangle are approximately 15.56 ft by 19.45 ft.
Case 3: Length to width ratio of 1 to 1.
For this case, the length is z and the width is z.
Perimeter = 70 ft
2 × (length + width) = 70
2 × (z + z) = 70
4z = 70
z = 70 / 4
z = 17.5
Length = z ≈ 17.5 ft
Width = z ≈ 17.5 ft
Therefore, for a perimeter of 70 ft and a length to width ratio of 1 to 1, the dimensions of the rectangle are approximately 17.5 ft by 17.5 ft.
The 1:1 case is obviously a square, with length and width both equal to 70/4.
Try the others yourself.
If (width)/(length) = 3/4,
L = (4/3)W
70 = 2(L + W) = (14/3)W
Solve for W