Question

The length of a rectangle is 8m less than thrice its width. if area of the rectangle is 1675 sq m them find the lenght and width of the rectangle​

Answer 1

Given:

The length of the rectangle is 8m less than thrice it's width.

So, let's assume length of the rectangle to be 3x - 8 m

Let the width be x m.

Area of the rectangle given as 1675 sq.m

Area of the rectangle:

Length × Breadth.

We will use the above formula to solve the question about the length and width of the rectangle.

So,

L × B = 1675

(3x - 8) x = 1675

Multiply the bracket (3x - 8) with x

3x² - 8x = 1675

Now form it in the form of an equation, in other words add a value to the RHS after shifting 1675 from RHS to LHS.

3x² - 8x - 1675 = 0

Now, comes the main task i.e to find two numbers whose product should be equal to a × c and and their sum up ( or on subtracting) should give the answer as middle term i.e b.

Compare the above equation:

3x² - 8x - 1675 = 0 with

ax² + bx + c = 0 and write the values of the variables.

So, we get

a = 3

b = 8

c = 1675

As mentioned above, find a number whose product should be equal to

a × c i.e (3 × 1675) and their sum up (or on subtracting) the answer should be b i.e the middle term (8)

So, the product of:

3 × 1675 = 5025

Now find two other numbers whose sum up or subtraction would give you answer the middle term (8)

To find such numbers I am also too bad at it but I can just suggest you to check the answer by multiplying numbers from 1 to 9 and then see for those pairs whose multiple gives you the last digit of the answer as 5 (only in this case)

For example: if we take the numbers 3 and 4 and multipy them the resultant number of their multiplication is 12 but our resultant number after multiplying is 5025 in which the last digit is 5 and not 2.

So, find two numbers within 1 to 9

whose resultant number's last digit would be 5.

So, let's try with 5 and 7 their multiplication gives us the answer as 35. Yes, we now have the last digit of the resultant number as 5.

So, now the thing is just to find two numbers whose product is 5025.

Let's try different combination of numbers, let's start from above 50, the first two number having the digit 5 and 7 at the end are:

55 × 57 = 3135

55 × 67 = 3685

Let's try other combinations of numbers.

67 × 55 = 3685

65 × 67 = 4355

75 × 77 = 5025

Yes, now we got two numbers whose product is equal to a × c and their sum up is b.

So let's move,

3x² - (75 - 67)x - 1675 = 0

3x² - 75x + 67x - 1675 = 0

3x ( x - 25) +67 ( x - 25) = 0

(x - 25) or (3x + 67) = 0

x - 25 or 3x + 67 = 0

x = 25 or 3x = - 67

x = 25 or x = - 67 / 3

Length cannot be negative

The width of the rectangle:

x = 25m

The length of the rectangle:

= 3 × 25 - 8

= 75 - 8

= 67

Therefore, the length of the rectangle = 67m

The width of the rectangle = 25m

Answer 2

Answer :-

→ Length = 67 m and breadth = 25 m.

Step-by-step explanation :-

First,

Let the width of rectangle be x.

And, then length = 3x - 8 .

And,

we have,

→ Area of rectangle = 1675 m² .

→ Length × width = 1675 .

→ ( 3x - 8 ) x = 1675 .

→ 3x² - 8x = 1675 .

→ 3x² - 8x - 1675 = 0 .

→ 3x² - 75x + 67x - 1675 = 0 .

→ 3x( x - 25 ) + 67( x - 25 ) = 0 .

→ ( 3x + 67 ) ( x - 25 ) = 0 .

→ x = 25 or -67/3 [ reject ; as length never be in negative ] .

$\therefore$ x = 25 m.

$\therefore$ width( x ) = 25 m.

And, length = 3x - 8 = 3(25) - 8 = 75 - 8 = 67 m.

Hence, it is solved.