A person has a rectangle or bore 12” x 40” around which she wants to put a uniform border of shells if she has enough shells for a border whose area is 360 in.² determine the width of the border
A person has a rectangle or bore 12” x 14” around which she wants to put a uniform border of shells if she has enough shells for a border whose area is 360 in.² determine the width of the border
AAAaannndd the bot gets it wrong yet again!
If the width of the border is w, then we have
(12+2w)(14+2w)-12*40 = 360
4w^2+52w-360 = 0
4(w+18)(w-5) = 0
w = 5
I apologize for the mistake in the previous answer. You are correct in your solution. Thank you for providing the correct method to solve the problem.
To determine the width of the border, we need to find the difference between the outer rectangle's area and the inner rectangle's area. Here are the steps:
1. Find the area of the outer rectangle:
Area = Length x Width
Area = 12" x 40"
Area = 480 in²
2. Find the area of the inner rectangle:
Area = Length x Width
Area = (12 - 2x) x (40 - 2x)
Area = (12x - 2x²) x (40x - 2x²)
Area = 480x - 84x² + 4x⁴
3. Set up an equation to find the width of the border:
480 - 84x² + 4x⁴ - 360 = 0
4. Rearrange the equation and simplify:
4x⁴ - 84x² + 120 = 0
5. Divide the equation by 4 to simplify further:
x⁴ - 21x² + 30 = 0
6. Solve the quadratic equation by factoring:
(x² - 6)(x² - 5) = 0
7. Set each factor equal to zero:
x² - 6 = 0
x² - 5 = 0
8. Solve for x:
x₁ = √6 ≈ 2.45
x₂ = √5 ≈ 2.24
9. Since we're dealing with dimensions, the border width cannot be negative. Therefore, we consider the positive value only.
The width of the border is approximately 2.45 inches.
To determine the width of the border, we first need to find the area of the rectangle and then subtract the area of the inner rectangle (12" x 40") to get the area of the border.
1. Calculate the area of the rectangle:
Area = length x width
Area = 12" x 40" = 480 in²
2. Subtract the area of the inner rectangle from the total area to get the area of the border:
Border Area = Total Area - Inner Rectangle Area
Border Area = 480 in² - (12 in x 40 in) = 480 in² - 480 in² = 0 in²
Since the border area is 0 in², it means there is no room for a border with the given dimensions. Therefore, there is no width for the border in this case.
We start by finding the area of the rectangle, which is:
12 x 40 = 480 in²
Next, we need to subtract the area of the original rectangle from the area that the rectangle plus the border will occupy, which is:
480 + 360 = 840 in²
840 - 480 = 360 in²
So the area of the border is 360 in².
We can set up the equation:
width x length = 360
We know that the length of the border will be the same as the length of the original rectangle, which is 40 inches. So we can solve for the width:
width x 40 = 360
width = 360/40
width = 9 inches
Therefore, the width of the border of shells will be 9 inches.
We start by finding the area of the rectangle, which is:
12 x 14 = 168 in²
Next, we need to subtract the area of the original rectangle from the area that the rectangle plus the border will occupy, which is:
168 + 360 = 528 in²
528 - 168 = 360 in²
So the area of the border is 360 in².
We can set up the equation:
width x length = 360
We know that the length of the border will be the same as the length of the original rectangle, which is 14 inches. So we can solve for the width:
width x 14 = 360
width = 360/14
width ≈ 25.71 inches (rounded to two decimal places)
Therefore, the width of the border of shells will be approximately 25.71 inches.