A rectangle has a length that is 1 less than twice its width. It's area is 21. Find its perimeter.
width --- x
length ---2x - 1
x(2x-1) = 21
2x^2 -x -21 = 0
((x+3)(2x-7)=0
x = -3 , silly or
x = 7/2 which is 3.5
width = 3.5
length = 6
perimeter = 2(3.5+6) = 19
A = lw
l = 2x-1
w = x
21 = x(2x -1)
21 = 2x^2 -x
2x^2-x -21 =0
(x +3)(2x -7)
x = 7/2 = 3.5
2(7/2)-1 = 6
l = 6; w = 7/2
p = 2l + 2w
p = 2(6) + 2(3.5) = 19
How did you get the value of x from (x+3)(2x-7)=0
To solve this problem, we'll first need to set up an equation based on the given information. Let's start by defining the width of the rectangle as 'w'.
Given that the length is 1 less than twice the width, we can express the length as (2w - 1).
The area of a rectangle is calculated by multiplying its length by its width. In this case, we know that the area is 21, so we can write the equation:
w * (2w - 1) = 21
Now, let's solve this equation to find the value of 'w'.
Expanding the equation, we get:
2w^2 - w - 21 = 0
Now we have a quadratic equation. To solve it, we can either factor it or use the quadratic formula. Let's solve it using the quadratic formula:
w = [-b ± √(b^2 - 4ac)] / 2a
In this equation, a = 2, b = -1, and c = -21. Plugging in these values, we get:
w = [1 ± √((-1)^2 - 4 * 2 * (-21))] / (2 * 2)
w = [1 ± √(1 + 168)] / 4
w = [1 ± √169] / 4
w = [1 ± 13] / 4
Now, we have two possible values for 'w':
w1 = (1 + 13) / 4 = 14 / 4 = 3.5
w2 = (1 - 13) / 4 = -12 / 4 = -3
Since widths cannot be negative, we discard w2 as an extraneous solution. Therefore, the width of the rectangle is 3.5.
Now, we can find the length using the expression 2w - 1:
length = 2 * 3.5 - 1 = 7 - 1 = 6
So, the length of the rectangle is 6.
To find the perimeter of a rectangle, you add up the lengths of all its sides. In this case, we have two equal sides (length and width), so we can use the formula: perimeter = 2 * (length + width).
Plugging in the values, we get:
perimeter = 2 * (6 + 3.5)
perimeter = 2 * 9.5
perimeter = 19
Therefore, the perimeter of the rectangle is 19 units.