Question

Represent the following situations in the form of quadratic equations:
(i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance.What is the speed of the train?
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Answer 1

Answer:

i)

Let the breadth be x m and the length will be 2x+1 m.

Area =l×b

Area =x(2x+1)=528

2x

2

+x−528=0

2x

2

+33x−32x−528=0

⇒2x(x−16)+33(x−16)=0

⇒(2x+33)(x−16)=0

⇒x=16,

2

33

Breadth =16 m and length =33 m

ii)

Let one number be x then the next number will be x+1

x(x+1)=306

⇒x

2

+x−306=0

⇒x

2

+18x−17x−306=0

⇒x(x+18)−17(x+18)=0

⇒(x+18)(x−17)=0

⇒x=17,−18

The numbers are 17 and 18.

iii)

Let Rohan's present age =x yrs.

Then his mother's present age =x+26 yrs

After 3 yrs

Rohan's age =x+3 yrs

His mother's age =x+29 yrs

(x+3)(x+29)=360

⇒x

2

+32x+87−360=0

⇒x

2

+32x−273=0

⇒x

2

+39x−7x−273=0

⇒x(x+39)−7(x+39)=0

⇒(x+39)(x−7)=0

⇒x=7,−39

So, Rohan's present age =7 yrs.

iv)

Let the speed of the train =x km/hr

(x−8)

480

−

x

480

=3

⇒480x−480x+3840=3x(x−8)

⇒3x

2

−24x−3840=0

⇒x

2

−8x−1280=0

⇒x

2

−40x+32x−1280=0

⇒x(x−40)+32(x−40)=0

⇒(x+32)(x−40)=0

⇒x=40,−32

The speed of the train is 40 km/hr.

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Answer 2

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(i) Let us consider,

The breadth of the rectangular plot is x m.

Thus, the length of the plot = (2x + 1) m

As we know,

Area of rectangle = length × breadth = 528 m2

Putting the value of length and breadth of the plot in the formula, we get,

(2x + 1) × x = 528

⇒ 2x2 + x = 528

⇒ 2x2 + x – 528 = 0

Hence, 2x2 + x – 528 = 0, is the required equation which represents the given situation.

(ii) Let us consider,

speed of train = x km/h

And

Time taken to travel 480 km = 480 (x) km/h

As per second situation, the speed of train = (x – 8) km/h

As given, the train will take 3 hours more to cover the same distance.

Therefore, time taken to travel 480 km = (480/x) + 3 km/h

As we know,

Speed × Time = Distance

Therefore,

(x – 8)[(480/x) + 3] = 480

⇒ 480 + 3x – (3840/x) – 24 = 480

⇒ 3x – (3840/x) = 24

⇒ 3x2 – 24x – 3840 = 0

⇒ x2 – 8x – 1280 = 0

Hence, x2 – 8x – 1280 = 0 is the required representation of the problem mathematically

Hope it's Helpful.....:)