The area of the larger rectangle is given by the expression x^2+9x+14 and the area of the smaller rectangle is given by the expression x^2+3x. Find the difference in the perimeters of the rectangles.
The area of a rectangle is given by length times width. In this case, the larger rectangle has a length of x+7 and a width of x+2, while the smaller rectangle has a length of x and a width of x+3.
So we have the equations:
(x+7)(x+2) = x^2+9x+14
x(x+3) = x^2+3x
Expanding the expressions, we have:
x^2+9x+14 = x^2+3x+9
Subtracting x^2 from both sides gives:
9x+14 = 3x+9
Subtracting 3x from both sides gives:
6x+14 = 9
Subtracting 14 from both sides gives:
6x = -5
Dividing both sides by 6 gives:
x = -5/6
The perimeters of the larger rectangle is given by:
2(x+7) + 2(x+2) = 2x+14+2x+4 = 4x+18
The perimeters of the smaller rectangle is given by:
2(x+3) + 2x = 2x+6+2x = 4x+6
Substituting x = -5/6 into both expressions, we have:
Perimeter of larger rectangle = 4(-5/6)+18 = -20/3+54/3 = 34/3
Perimeter of smaller rectangle = 4(-5/6)+6 = -20/3+18/3 = -2/3
The difference in the perimeters is: 34/3 - (-2/3) = 34/3 + 2/3 = 36/3 = 12
Therefore, the difference in the perimeters of the rectangles is 12.
Can you explain all the steps
Certainly! Let's go through the steps one by one:
1. We start by setting up the area equations for both rectangles. The larger rectangle has a length of x+7 and a width of x+2, so its area is given by (x+7)(x+2). The smaller rectangle has a length of x and a width of x+3, so its area is given by x(x+3).
2. We expand the expressions on both sides of the equations:
(x+7)(x+2) = x^2 + 9x + 14
x(x+3) = x^2 + 3x
3. We set the two expanded expressions equal to each other, since the areas of the rectangles are equal.
x^2 + 9x + 14 = x^2 + 3x
4. We cancel out the x^2 terms on both sides by subtracting x^2 from both sides:
9x + 14 = 3x
5. We move all the x terms to one side by subtracting 3x from both sides:
9x - 3x + 14 = 0
6. We simplify the equation:
6x + 14 = 0
7. We move the constant term to the other side by subtracting 14 from both sides:
6x = -14
8. We solve for x by dividing both sides by 6:
x = -14/6 = -7/3
9. Now we know the value of x, so we can substitute it back into the expressions for the perimeters. The formula for the perimeter of a rectangle is 2(length + width).
10. For the larger rectangle, the length is x+7 and the width is x+2. So the perimeter of the larger rectangle is:
2(x+7) + 2(x+2) = 2x + 14 + 2x + 4 = 4x + 18
11. For the smaller rectangle, the length is x and the width is x+3. So the perimeter of the smaller rectangle is:
2(x+3) + 2x = 2x + 6 + 2x = 4x + 6
12. We substitute x = -7/3 into both expressions to find the specific perimeters:
Perimeter of larger rectangle = 4(-7/3)+18 = -28/3 + 18 = -28/3 + 54/3 = 26/3
Perimeter of smaller rectangle = 4(-7/3)+6 = -28/3 + 18/3 = -10/3
13. Finally, we find the difference in the perimeters by subtracting the perimeter of the smaller rectangle from the perimeter of the larger rectangle:
Difference in perimeters = 26/3 - (-10/3) = 26/3 + 10/3 = 36/3 = 12
Therefore, the difference in the perimeters of the rectangles is 12.