a rectangular park is to be …

Let breadth of the rectangular park = {tex}x {/tex} m

Then, length of the rectangular park = ({tex}x+3{/tex}) m

Now, area of the rectangular park is = {tex}x(x+3)=(x^2+3x)m^2{/tex} {tex}[\because area=length\times breadth]{/tex}

Given, base of the triangular park = Breadth of the rectangular park

Therefore base of triangular park is = {tex}x{/tex} m

and it is given that altitude of triangular park is = {tex}12 {/tex} m

Therefore, area of the triangular park will be = {tex}\frac{1}{2}\times x\times 12= 6x {/tex} {tex}m^2{/tex} {tex}[\because area(Triangle)=\frac{1}{2}\times base\times altitude]{/tex}

As per the question area of rectangular park is = {tex}4 + {/tex} Area of triangular park

{tex}\Rightarrow x^2+3x=4+6x{/tex}

{tex}\Rightarrow x^2+3x-6x-4=0{/tex}

{tex}\Rightarrow x^2-3x-4=0{/tex}

{tex}\Rightarrow x^2-4x+x-4=0{/tex} [ by factorization ]

{tex}\Rightarrow x(x-4)+1(x-4)=0{/tex}
{tex}\Rightarrow (x-4)(x+1)=0{/tex}

{tex}\Rightarrow x-4=0 {/tex} or {tex}x+1=0{/tex} {tex}{/tex}
⇒ x = 4 or x = - 1

Since, breadth cannot be negative , so we will neglect {tex}x=-1{/tex} and choose {tex}x = 4{/tex}

Hence, breadth of the rectangular park will be = 4 m

and length of the rectangular park  will be = {tex}x+3=4+3=7m{/tex}

Verification:

Area of rectangular park is = 28 m²

Area of triangular park is = 24 m² = (28 – 4) m²