A rectangle has dimensions of 12x8inches. If a uniform border is added, then the new rectangle has twice the area of the old rectangle.
a) Find the width of the border
b) find the new dimensions of the rectangle
Call the border width w. The bordered rectangle dimensions are (12+w) x (8+w)
Solve the equation
(12+w)(8+w)= 2*12*8 = 192
96 + 20w + w^2 = 192
w^2 + 20w - 96 = 0
(w+24)(w-4) = 0
w = 4 or -24
Take the positive root for w. Add it to the length and the width for the new dimensions.
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sample
To find the width of the border, we can set up an equation based on the information given. Let's denote the width of the border as "x".
The original rectangle has dimensions 12x8 inches. When a uniform border is added, the new dimensions become (12 + 2x) x (8 + 2x) inches.
We are given that the new rectangle has twice the area of the old rectangle. So, we can set up the equation:
(12 + 2x) * (8 + 2x) = 2 * (12 * 8)
To solve this equation, we can simplify the right side of the equation:
(12 + 2x) * (8 + 2x) = 2 * 96
(12 + 2x) * (8 + 2x) = 192
Expanding both sides of the equation gives us:
96 + 20x + 4x^2 = 192
Rearranging the terms and simplifying further:
4x^2 + 20x - 96 = 0
We can factor or use the quadratic formula to solve this quadratic equation. After solving, we find that x = 3.
Therefore, the width of the border is 3 inches (a).
To find the new dimensions of the rectangle (b), we substitute the width of the border (x=3) back into the expression for the new dimensions:
New width = 12 + 2x = 12 + 2(3) = 12 + 6 = 18 inches
New height = 8 + 2x = 8 + 2(3) = 8 + 6 = 14 inches
Hence, the new dimensions of the rectangle are 18 x 14 inches.