Question

A rectangular park is to be designed whose breath is 3 m less than its length. Its area to be 4 sq m more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12 metre. Find its length and breadth of the rectangular Park.​

Answer 1

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⇒ breadth of the rectangular park = 4m

⇒ length of the rectangular park = 7m

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$\tt\red {GIVEN :}$

Breadth of the rectangular park is 3 m less than its length and area of rectangular park is 4m² less than the area of the isosceles triangle and altitude of the triangular park is 12 m.

$\tt\orange {TO \: FIND :}$

Length and breadth of the rectangular park.

$\tt\green {FORMULA \: USED :}$

Area of rectangular park = l × b

where,

L = length of the rectangular park

B = breadth of the rectangular park

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Area of triangular park = 1/2 × b × h

where,

B = base of the triangular park

H = altitude of the triangular park

$\tt\blue { CALCULATIONS :}$

Let breadth of rectangular park = x m

Then, length of rectangular park = (x + 3) m

Now, area of rectangular park = l × b

$\tt \implies \: x(x + 3) = (x² + 3x)m²$

Given, base of triangular park = breadth of rectangular park

∴Base of triangular park = xm

and also it is given that altitude of triangular park = 12m

∴Area of triangular park = $\tt \frac{1}{2} \times b \times h$

substituting the values,

$\tt \implies \: \frac{1}{2} × x × 12 = 6x m²$

According to the question,

Area of rectangular park = 4 + Area of triangular park

$\tt\implies x² + 3x = 4 + 6x$

$\tt\implies x² + 3x - 6x - 4 = 0$

$\tt\implies x² - 3x - 4 = 0$

$\tt\implies x² - 4x + x - 4 = 0 \: \: \: \: \: .. (By \: factorization)$

$\tt\implies x(x + 4) +1(x - 4) = 0$

$\tt\implies (x - 4) (x + 1) = 0$

$\tt\implies x - 4 = 0$

$\tt\implies x = 4$

and

$\tt\implies x + 1 = 0$

$\tt\implies x = -1$

Since, breadth cannot be negative, so neglect x = -1

∴ x = 4

Hence, breadth of the rectangular park = 4m

Hence, breadth of the rectangular park = 4mand length of the rectangular park = x + 3 = 4 + 3 = 7m

Answer 2

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refer to the attachment :)