A rectangle has a perimeter of 114m and an area of 807m^2. What are the dimensions?
To find the dimensions of the rectangle, we need to set up a system of equations based on the given information.
Let's denote the length of the rectangle as 'l' and the width as 'w'.
We know that the perimeter of a rectangle is given by the formula: P = 2(l + w).
Given that the perimeter is 114m, we can write the equation as: 2(l + w) = 114.
Next, we know that the area of a rectangle is given by the formula: A = l * w.
Given that the area is 807 square meters, we can write the equation as: l * w = 807.
Now, we have a system of equations:
1. 2(l + w) = 114,
2. l * w = 807.
To solve this system, we can use the substitution method.
Let's solve equation 1 for l:
2(l + w) = 114
=> l + w = 57
=> l = 57 - w
Now substitute the value of l in equation 2:
(57 - w) * w = 807.
Expanding and simplifying this equation produces a quadratic equation:
57w - w^2 = 807.
Rearranging the equation to standard form:
w^2 - 57w + 807 = 0.
To solve this quadratic equation, we can factor it or use the quadratic formula.
Factoring might not be straightforward in this case, so let's use the quadratic formula, where a = 1, b = -57, and c = 807:
w = (-b ± √(b^2 - 4ac)) / (2a).
Substituting the values in the formula gives:
w = (-(-57) ± √((-57)^2 - 4(1)(807))) / (2(1))
= (57 ± √(3249 - 3228)) / 2
= (57 ± √21) / 2.
Therefore, the two possible values for w are:
w₁ = (57 + √21) / 2,
w₂ = (57 - √21) / 2.
Now we can substitute each value of w into equation 1 to find the corresponding value of l.
For w₁:
l = 57 - w₁
= 57 - [(57 + √21) / 2]
= 57 - (57/2) - (√21/2)
= (114/2) - (√21/2)
= 57 - (√21/2).
For w₂:
l = 57 - w₂
= 57 - [(57 - √21) / 2]
= 57 - (57/2) + (√21/2)
= (114/2) + (√21/2)
= 57 + (√21/2).
Therefore, the dimensions of the rectangle are:
w₁ = (57 + √21) / 2 meters (width)
l₁ = 57 - (√21/2) meters (length)
w₂ = (57 - √21) / 2 meters (width)
l₂ = 57 + (√21/2) meters (length)