The area of certain rectangle is 288 yd. the perimeter is 68 yd. if you double the length and width, what will be the area and perimeter of the new rectangle.
To find the dimensions of the rectangle, we can set up a system of equations based on the given information.
Let's say the length of the rectangle is "L" and the width is "W".
From the given information, we have two equations:
1. The area of the rectangle: L × W = 288
2. The perimeter of the rectangle: 2L + 2W = 68
To solve these equations, we can use substitution or elimination methods. In this case, let's solve by substitution.
From equation 1, we can express L in terms of W:
L = 288 ÷ W
Now substitute this value of L in equation 2:
2(288 ÷ W) + 2W = 68
Simplify the equation:
576 ÷ W + 2W = 68
Multiply both sides by W:
576 + 2W^2 = 68W
Rearrange the equation:
2W^2 - 68W + 576 = 0
Now we need to solve this quadratic equation to find the possible values of W, which will give us the dimensions of the rectangle.
Using factoring, completing the square, or the quadratic formula, we find that W = 8 or W = 36.
If W = 8, then L = 288 ÷ 8 = 36.
If W = 36, then L = 288 ÷ 36 = 8.
So the two possible sets of dimensions for the original rectangle are 8 yards by 36 yards or 36 yards by 8 yards.
Now we can double the length and width to find the new rectangle's dimensions:
For the first set of dimensions (8 yards by 36 yards):
Length: 2 × 8 = 16 yds
Width: 2 × 36 = 72 yds
For the second set of dimensions (36 yards by 8 yards):
Length: 2 × 36 = 72 yds
Width: 2 × 8 = 16 yds
Now we can find the new area and perimeter of each resulting rectangle.
For the first set of dimensions (16 yards by 72 yards):
New area: 16 × 72 = 1152 yd^2
New perimeter: 2(16) + 2(72) = 32 + 144 = 176 yd
For the second set of dimensions (72 yards by 16 yards):
New area: 72 × 16 = 1152 yd^2
New perimeter: 2(72) + 2(16) = 144 + 32 = 176 yd
So, regardless of the order of length and width, the new rectangle will have an area of 1152 square yards and a perimeter of 176 yards.
L * W = 288
W = 288/L
2L + 2W = 68
Substitute 288/L for W in last equation and solve for L. Insert that value into the first equation and solve for W. Check by inserting both values into the last equation.