Question

a rectangular field is 90 yards long and 60 yards wide. give the length and width of another rectangular field that has the same perimeter but a larger area. width = yards length = yards

Explanation:

Step1: Calculate the perimeter of the original rectangle

The perimeter formula for a rectangle is \(P = 2(l + w)\). For \(l = 90\) yards and \(w = 60\) yards, \(P=2(90 + 60)=2\times150 = 300\) yards.

Step2: Recall the relationship between area and side - length for a rectangle

The area formula for a rectangle is \(A=l\times w\), and from the perimeter \(2(l + w)=300\), we have \(l + w=150\), so \(l = 150 - w\). Then \(A=(150 - w)w=-w^{2}+150w\). This is a quadratic function \(y = ax^{2}+bx + c\) (\(a=-1\), \(b = 150\), \(c = 0\)). The maximum of a quadratic function \(y=ax^{2}+bx + c\) occurs at \(x=-\frac{b}{2a}\). Here, \(w=-\frac{150}{2\times(-1)} = 75\) (when the rectangle is a square, the area is maximized for a given perimeter). But we can choose values close to the square - like dimensions. Let's take \(w = 70\) yards.

Step3: Calculate the length

Since \(l+w = 150\), when \(w = 70\) yards, \(l=150 - 70=80\) yards.

Step4: Calculate the areas

Original area: \(A_{1}=90\times60 = 5400\) square yards. New area: \(A_{2}=80\times70=5600\) square yards.

Answer:

width = \(70\) yards, length = \(80\) yards