Question

a rectangular park is to be designed whose breadth is 3m less than alength . its area is to be 4 square metres more than area of park that has already been made in the shape of an isciolces triangle with its base as the breadth of the rectangular park and of alititude 12m..... find its length and breadth

Answer 1

let the length be l
then breadth= l-3
area of rectangle= l(l-3)= l2-3l
area of triangle= (1/2)(l-3)(12)= 6l-18
given
area of rectangle is 4sq mt more than triangle
so
area of rectangle -4= area of triangle
l2-3l -4=6l-18
l2-9l+14= 0
(l-7)(l-2)= 0
l= 7....... and 2
l=2 is neglected as when length is 2 the breadth will be negative which is not possible...
so length= 7m
breadth= l-3=7-3=4m

Answer 2

Let the length of the rectangular park = x m

Then, it's length = (x+3)m

$\therefore$ Area of the rectangular park =

$\sf{x(x + 3) = ( {x}^{3} + 3x) {m}^{2} }$

Now, base of the isosceles triangle = x m

$\therefore$ It's area = $\sf{\frac{1}{2}×x×12=6x{m}^{2}}$

A/C,

Area of rectangular park = Area of the triangular park + $\sf{4{{m}^{2}}$

$\longrightarrow$$\sf{ {x}^{2} + 3x = 6x + 4}$

$\longrightarrow$$\sf{ {x}^{2} - 3x - 4 = 0}$

$\longrightarrow$$\sf{{x}^{2} - 4x + x - x = 0}$

$\longrightarrow$$\sf{(x + 1)(x - 4) = 0}$

$\longrightarrow$$\sf{x = - 1 \: or \: x = 4}$

As the breadth of the park cannot be negative, x ≠ -1, x = 4

Hence, the breadth of the park = 4m

And it's length = 4 + 3 = 7m

Verification

Area of rectangular park = 4m × 7m = $\sf{28{m}^{2}}$

Area of triangular park = $\sf{ \frac{1}{2} \times 4 \times 12 = 24 {m}^{2} = (28 - 4) {m}^{2}}$