The area of a rectangle is 84 square units. If its length is shortened by 7 units and its width by 2 units a square is formed. Determine the area of the square.
Let s be the square.
lw=84
(l-7)=s=(w-2)
l= w+5
(w+5)w=84
multiply that out, solve for w (it is a quadratic).
Let's assume the length of the rectangle is L units and the width is W units.
We are given that the area of the rectangle is 84 square units, so we can write the equation:
L * W = 84
We are also given that if we decrease the length by 7 units and the width by 2 units, a square is formed. This means the new length and width are equal, so we can write:
(L - 7) * (W - 2) = (L - 7)²
To find the area of the square, we need to determine the value of (L - 7)².
Let's solve the first equation for L in terms of W:
L = 84 / W
Substituting this into the second equation, we get:
(84 / W - 7) * (W - 2) = (84 / W - 7)²
Expanding both sides of the equation, we have:
(84 - 7W / W) * (W - 2) = (84 - 7W / W)²
Simplifying further:
(84 - 7W) / W * (W - 2) = (84 - 7W)² / W²
Cross-multiplying, we get:
(84 - 7W) * (W - 2) * W² = (84 - 7W)²
Expanding and simplifying, we have:
84W³ - 168W² - 588W + 1176 = 0
Now we need to solve this cubic equation for W to find the width of the rectangle. We can use numerical methods or a graphing calculator to find the approximate values of W.
Let's assume one of the values of W is W₁.
Substituting this value back into the first equation to find the corresponding length:
L₁ = 84 / W₁
Now, we can calculate the area of the square using the formula:
Area of square = side length²
Area of square = (L₁ - 7)²
Repeat the same steps for the other possible values of W to find the corresponding lengths and areas of the square.
Note: There may be multiple possible solutions, so be sure to check all the values of W and find the corresponding areas of the square for each case.
To solve this problem, we need to determine the length and width of the rectangle, and then use that information to find the area of the square.
Let's consider the rectangle's area, which is given as 84 square units. We know that the formula for the area of a rectangle is length multiplied by width. So, we can express the area as:
Length * Width = 84
Now, we are given that if we decrease the length by 7 units and the width by 2 units, a square is formed. This means the length and width of the square are equal.
Let's assume the length of the rectangle is L and the width is W.
According to the given condition, the length of the square (L - 7) is equal to the width of the square (W - 2):
L - 7 = W - 2
Simplifying the equation, we have:
L - W = 5
Now, we have two equations:
1) Length * Width = 84
2) Length - Width = 5
We can solve these equations simultaneously to find the values of length (L) and width (W). Let's use the substitution method.
From equation 2, we have:
Length = Width + 5 (equation 3)
Substituting equation 3 into equation 1, we get:
(Width + 5) * Width = 84
Simplifying further:
Width^2 + 5Width - 84 = 0
Now, we can solve this quadratic equation to find the possible values of Width. By factoring or using the quadratic formula, we find that Width = 7 or Width = -12 (but width can't be negative in this context).
So, the width of the rectangle is 7 units. Substituting this back into equation 3, we can find the length:
Length = 7 + 5 = 12 units
Now, we have the dimensions of the rectangle, which are Length = 12 units and Width = 7 units.
To determine the area of the square, we need to find the length of one side. Since the length and width of the square are equal, we can use either dimension. Let's use the length.
Length of the square = Length of the rectangle - 7 = 12 - 7 = 5 units
The area of a square is given by side length squared. So, let's calculate the area of the square:
Area of the square = (Side length)^2 = 5^2 = 25 square units
Therefore, the area of the square is 25 square units.