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Q2(iv):
Represent the following situations in the form of quadratic equations : (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.
Solution :
Given:
- Total distance to be covered by the train, $d = 480$ km.
- The train travels at a uniform speed.
- If the speed were $8$ km/h less, the time taken to cover the same distance would increase by $3$ hours.
To Find:
Represent the given situation in the form of a quadratic equation in terms of the speed of the train.
Step 1: Defining Variables
Let the uniform speed of the train be $x$ km/h.
We know the fundamental relationship between distance, speed, and time is given by the formula:
$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$
Step 2: Formulating Time Expressions
Case 1: When the train travels at the original uniform speed $x$ km/h.
Time taken ($t_1$) = $\frac{480}{x}$ hours.
Case 2: When the speed is reduced by $8$ km/h.
New speed = $(x - 8)$ km/h.
Time taken ($t_2$) = $\frac{480}{x - 8}$ hours.
Step 3: Establishing the Relationship
According to the problem, the train takes $3$ hours more when the speed is reduced. Therefore, the difference between the new time ($t_2$) and the original time ($t_1$) is $3$ hours:
$t_2 - t_1 = 3$
Substituting the expressions from Step 2:
$\frac{480}{x - 8} - \frac{480}{x} = 3$
Step 4: Simplifying the Equation
Divide the entire equation by $3$ to simplify the coefficients:
$\frac{160}{x - 8} - \frac{160}{x} = 1$
Find a common denominator for the left-hand side, which is $x(x - 8)$:
$\frac{160x - 160(x - 8)}{x(x - 8)} = 1$
Expand the numerator:
$\frac{160x - 160x + 1280}{x^2 - 8x} = 1$
Simplify the numerator ($160x - 160x = 0$):
$\frac{1280}{x^2 - 8x} = 1$
Step 5: Final Algebraic Form
Multiply both sides by $(x^2 - 8x)$:
$1280 = x^2 - 8x$
Rearrange the terms to the standard form of a quadratic equation $ax^2 + bx + c = 0$:
$x^2 - 8x - 1280 = 0$
Final Answer: The quadratic equation representing the situation is $x^2 - 8x - 1280 = 0$, where $x$ is the speed of the train in km/h.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.1
- Q1(i): Check whether the following are quadratic equations : (i) $(x + 1)^2 = 2(x – 3)$
- Q1(ii): Check whether the following are quadratic equations : (ii) $x^2 – 2x = (–2)(3 – x)$
- Q1(iii): Check whether the following are quadratic equations : (iii) $(x – 2)(x + 1) = (x – 1)(x + 3)$
- Q1(iv): Check whether the following are quadratic equations : (iv) $(x – 3)(2x +1) = x(x + 5)$
- Q1(v): Check whether the following are quadratic equations : (v) $(2x – 1)(x – 3) = (x + 5)(x – 1)$
- Q1(vi): Check whether the following are quadratic equations : (vi) $x^2 + 3x + 1 = (x – 2)^2$
- Q1(vii): Check whether the following are quadratic equations : (vii) $(x + 2)^3 = 2x (x^2 – 1)$
- Q1(viii): Check whether the following are quadratic equations : (viii) $x^3 – 4x^2 – x + 1 = (x – 2)^3$
- Q2(i): Represent the following situations in the form of quadratic equations : (i) The area of a rectangular plot is $528$ $m^2$. 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.
- Q2(ii): Represent the following situations in the form of quadratic equations : (ii) The product of two consecutive positive integers is $306$. We need to find the integers.
- Q2(iii): Represent the following situations in the form of quadratic equations : (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.
CBSE Solutions for Class 10 Mathematics Quadratic Equations
Chapters in CBSE - Class 10 Mathematics
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