Problem 1 :
The area of a rectangular plot is 528 m^{2}. The length of the plot (in meters) is one more than twice its width. Find the length and width of the plot.
Solution :
Let x be the width of the plot.
Then the length of the plot is
= 2x + 1
Area of rectangular plot = 528 m^{2}.
length ⋅ width = 528
(2x + 1)x = 528
2x^{2} + x = 528
2x^{2} + x - 528 = 0
Solving the above quadratic equation, we get
x = 16 or x = -16.5
Since x represents width of the rectangle, it can not have a negative value. So, we can ignore x = -16.5.
Therefore, x = 16.
Length = 16 m
Width = 2(16) + 1
= 32 + 1
= 33 m
Problem 2 :
If the product of two consecutive positive integers is 306. then find the integers.
Solution :
Let x and x + 1 be the two consecutive positive integers.
Product of these integers = 306
x(x + 1) = 306
x^{2} + x = 306
x^{2} + x - 306 = 0
Solving the above quadratic equation, we get
x = 17 or x = -18
Since x is a positive integer, we can ignore x = -18.
x = 18
x + 1 = 18 + 1
= 19
Therefore, the two consecutive positive integers are 18 and 19.
Problem 3 :
Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. Find the present age of Rohan.
Solution :
Let x be Rohan's present age.
Then, the present age of Rohan's mother is
= x + 26
Three years later,
Rohan's age = x + 3
Mother's age = x + 26 + 3
Product of their ages = 360
(x + 3)(x + 29) = 360
x^{2} + 29x + 3x + 87 = 360
x^{2} + 32x + 87 = 360
x^{2} + 32x + 87 - 360 = 0
x^{2} + 32x - 273 = 0
Solving the above quadratic equation, we get
x = 7 or x = -39
Since x represents the present age of Rohan, we can ignore x = -39.
Therefore, the present age of Rohan is 7 years.
Problem 4 :
A train travels 480 km of distance at a uniform speed. If the speed is reduced by 8 km/hr, then it takes 3 hours more to cover the same distance. Find the speed of the train.
Solution :
Let x be the original speed of the train.
Time = Distance/Speed
Time taken by the train to cover 480 km distance at the speed of x km/hr :
= 480/x
Time taken by the train to cover 480 km distance in the reduced speed, that is (x - 8) km/hr :
= 480/(x - 8)
Given : When the speed is (x - 8) km/hr, it takes 3 hours more to cover the same distance.
480/(x - 8) = (480/x) + 3
480/(x - 8) = (480 + 3x)/x
480x = (3x + 480)(x - 8)
480x = 3x^{2} - 24x + 480x - 3840
Subtract 480x from both sides.
0 = 3x^{2} - 24x - 3840
Divide both sides by 3.
0 = x^{2} - 8x - 1280
Solving the above quadratic equation, we get
x = 40 or x = -32
Since x represents the original speed of the train, it can not be negative. So, we can ignore x = -32.
Therefore, original speed of the train is 40 km/hr.
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