Question

Represent the following situations in the form of quadratic equations.

(i) The area of a rectangular plot is 528m^2. 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) The product of two consecutive positive integers is 306. We need to find the integers.

(iii) Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan's present age.

(iv) 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. We need to find the speed of the train.​

Answer 1

$\huge \underline{ \blue{ \boxed{ \bf \red{Answer:-}}}}$

I'll give you full ANSWER :

(i) Let the breadth =b m

length = l =(2b+1)m

area =528m²

l*b= 528

(2b+1)b-528=0

2b²+b-528=0

2b²+33b-32b-528=0

b(2b+33)- 16(2b+33)=0

(2b+33)(b-16)=0

2b+33=0 or b-16=0

b should not be zero

therefore,

b-16=0

b=16

breadth = 16m

length =l= 2b+1=2*16+1=32+1=33m.

(ii) The product of two consecutive positive integers is 306.

We need to find the integers

solution : Let two consecutive numbers are x and (x + 1)

A/C to question,

product of x and (x + 1) = 306

⇒x(x + 1) = 306

⇒x² + x - 306 = 0

⇒ x² + 18x - 17x - 306 = 0

⇒x(x + 18) - 17(x + 18) = 0

⇒(x + 18)(x - 17) = 0⇒ x = 17 and -18

so x = 17 and (x +1) = 18

Therefore the numbers are 17 and 18.

(iii) Let the age of Rohan be “ x ”

Then Age of Rohan's mother = x+26

After three years their ages are (x+3),(x+29) respectively .

According to the question ;

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

x²+(3+29)x+87=360

x²+32x-273=0

x²+39x-7x-273=0

x(x+39)-7(x+39)=0

(x+39)(x-7)=0

x = 7 or -39

As Age can never be negative we have x =7.

Therefore ,Present age of Rohan =7 years.

(iv) The original speed of the train is 40 km/hour

Step-by-step explanation:

Let the original speed of the train be x km/hour

Distance traveled by the train = 480 km

\text{Time taken by train = }\frac{480}{x}\\\\\text{If the train would have taken 3 hours more then time = }\frac{480}{x-8}\\\\\implies \frac{480}{x}=\frac{480}{x-8}+3\\\\\implies \frac{480}{x}-\frac{480}{x-8}=3\\\\\implies 3x^2-24x+3840=0\\\\\implies x^2-8x+1280=0\\\\\implies \text{ x = -32 or x = 40}\\\\\text{But speed can not be negative }\implies x=40

Hence, the original speed of the train is 40 km/hour.

$\bf\blue{Hope\ it\ helps.}$

$\bf\pink{Plz\ Mark\ As\ Brainliest.}$

Answer 2

(i) Let the breadth of the plot be x m.

Hence, the length of the plot is (2x + 1) m.

Area of a rectangle Length x Breadth

528 = x (2x + 1)

$\tt: \implies2 {x}^{2} + x - 528 = 0$

(ii) Let the consecutive integers be x and x + 1.

It is given that their product is 306.

$\tt: \implies x(x + 1) = 306$

$\tt: \implies {x}^{2} + x - 306 = 0$

(iii) Let Rohan's age be x.

Hence, his mother's age = x + 26

3 years hence,

Rohan's age = x + 3

Mother's age = x + 26 + 3 = x + 29

It is given that the product of their ages after 3 years is 360

$\tt: \implies (x + 3) (x + 29) = 360$

$\tt: \implies {x}^{2} + 32x - 273 = 0$

(iv) Let the speed of train be x km/h.

Time taken to travel 480 km

$\tt: \implies \frac{480}{x} \: hrs$

In second condition, let the speed of train = (x-8) km/h

It is also given that the train will take 3 hours cover the same distance.

Therefore, time taken to travel 480 km

$\tt: \implies ( \frac{480}{x} - 3)$

hrs

Speed x Time Distance

$\tt : \implies (x + 3)( \frac{480}{x} + 3) = 480$

$\tt: \implies 480 + 3 - \frac{3840}{x} - 24 = 480$

$\tt: \implies 3x - \frac{3840}{x} = 24$

$\tt: \implies3 {x}^{2} - 24x - 3840 = 0$

$\tt: \implies {x}^{2} - 8x - 1280 = 0$