Question

A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square metres 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 m. Find the length and breadth of the rectangular park

Answer 1

Answer:

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Step-by-step explanation:

Here, Length of Rectangular Park = L

Breadth of Rectangular Park = L - 3

Altitude of Triangular Park = 12

Breadth of Triangular Park = L - 4

Now,

Area of rectangle =L (L - 3)

Area of triangle = 1/2 × 12(L - 3)

Area of rectangle - Area of triangle =4

⇒L (L - 3) - 1/2 × 12(L - 3) =4

⇒L² - 9L + 14 = 0

⇒(L - 2) (L - 7) = 0

⇒L = 2,7

Since, breadth is L - 3 , L ≠ 2

∴ L=7m & breadth =7 - 3 = 4m

Answer 2

$\overline{\underline{\boxed{\sf GIVEN \darr}}}$

A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square metres 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 m. Find the length and breadth of the rectangular park

$\overline{\underline{\boxed{\sf HOW TO SOLVE \darr}}}$

Step 1 : Let us assume length with any variable

Step 2 : Then as it is given breadth will the 3 subtracted from that variable.

Step 3 : Similarly, the same with the breadth of the triangular park.

Step 4 : Then Area of rect. - area of ∆ = 4

Step 5 : Then putting the values we will simplify

Step 6 : After simplifying we need to apply middle term factorisation from the chapter factorisation.

Step 7 : Then we will get two values 7 and 2

Step 8 : 2 will be ignored as length and breadth cannot be negative

Step 9 : Putting 7 in place of the variable we will get the values.

$\overline{\underline{\boxed{\sf SOLUTION \darr}}}$

Let us assume the length be x

Length of the rectangular park = x

Breadth of the rectangular park = x - 3

Area of the rectangular park is 12 m

Breadth of the triangular park = x - 3

Now,

$\frak{ \red{Area_{Rectangle} = x(x - 3)}}$

$\frak{ \purple{Area_{Rectangle} = \frac{1}{2} \times 12(x - 3) }}$

$\frak{ \orange{Area_{Rectangle} - Area_{Triangle} = 4}}$

$\leadsto\frak{ \green{x(x - 3) - \frac{1}{2} \times 12(x - 3) = 4 }}$

$\looparrowright\frak{ \pink{ {x}^{2} - 9x + 14 = 0 }}$

Now by middle term factorisation we get

$\leadsto \frak{ \gray{{x}^{2} - (2 + 7)x + 14 = 0}}$

$\looparrowright \frak{ \blue{(x - 2)(x - 7) = 0}}$

Now,

x - 2 = 0

• x = 2

x - 7 = 0

x = 7

And we know that breadth is x - 3 and if 2 is L so the it will be - 1 and breadth cannot be negative.

So,

x = 7 whereas x ≠ 2

So the length is 7m

Breadth = x - 3 = 7 - 3 → 4m