Question

a rectangular park is 110 yards long and 80 yards wide. give the length and width of another rectangular park that has the same perimeter but a larger area. width = yards length = yards

Explanation:

Step1: Calculate the perimeter of the original park

The perimeter formula for a rectangle is \(P = 2(l + w)\). For the given park with \(l = 110\) yards and \(w = 80\) yards, \(P=2(110 + 80)=2\times190 = 380\) yards.

Step2: Calculate the area of the original park

The area formula for a rectangle is \(A=l\times w\). So, \(A = 110\times80=8800\) square - yards.

Step3: Find new length and width

Let the new length be \(l_1\) and new width be \(w_1\) such that \(2(l_1 + w_1)=380\), so \(l_1 + w_1=190\), or \(w_1 = 190 - l_1\). The area \(A_1=l_1\times w_1=l_1(190 - l_1)=190l_1 - l_1^{2}\).
We know that for a quadratic function \(y = ax^{2}+bx + c\) (\(a=- 1\), \(b = 190\), \(c = 0\)), the vertex of the parabola \(y=-x^{2}+190x\) (since \(x=-\frac{b}{2a}\)) gives the maximum value of the area. But we can also use a trial - and - error method.
If we take \(l_1 = 100\) yards, then \(w_1=190 - 100 = 90\) yards.
The area \(A_1=100\times90 = 9000\) square - yards.

Answer:

width \( = 90\) yards, length \(=100\) yards