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Consider the equations given below:
y = (1 - x)2
y = 0
x = 0
A straight line representing x = k separates the area enclosed by the above curves. Say both the areas are A1 (0 ≤ x ≤ k) and A2 (k ≤ x ≤ 1). If A1 - A2 = 1/4 , then what is the value of k?
y = (1 - x)2
y = 0
x = 0
A straight line representing x = k separates the area enclosed by the above curves. Say both the areas are A1 (0 ≤ x ≤ k) and A2 (k ≤ x ≤ 1). If A1 - A2 = 1/4 , then what is the value of k?
- a)1/2
- b)1/4
- c)2/3
- d)1/3
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Consider the equations given below:y = (1 - x)2y = 0x = 0A straight li...
Here, area between 0 and k is A1, and between k and 1 is A2. Therefore,
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Community Answer
Consider the equations given below:y = (1 - x)2y = 0x = 0A straight li...
Understanding the Problem:
The given equations represent a parabola and a line. We need to find the value of k such that the area enclosed by the parabola and the line is 1/4.
Solution:
To find the value of k, we first need to determine the points of intersection of the parabola and the line x = k. Setting the two equations equal to each other:
(1 - k)^2 = 0
Solving for k, we get k = 1.
Now, we need to find the area enclosed by the parabola and the line x = 1. This can be done by integrating the parabola equation with respect to x from 0 to 1:
A1 = ∫[0,1] (1 - x)^2 dx
A1 = [x - (x^2)/2 + (x^3)/3] [0,1]
A1 = 1/3
Since A1 - A2 = 1/4, we have:
1/3 - A2 = 1/4
A2 = 1/3 - 1/4
A2 = 1/12
The area A2 is the area enclosed by the parabola and the line x = 1. Therefore, the value of k is such that A2 = 1/12, which corresponds to k = 1/2.
Therefore, the correct answer is option A) 1/2.
The given equations represent a parabola and a line. We need to find the value of k such that the area enclosed by the parabola and the line is 1/4.
Solution:
To find the value of k, we first need to determine the points of intersection of the parabola and the line x = k. Setting the two equations equal to each other:
(1 - k)^2 = 0
Solving for k, we get k = 1.
Now, we need to find the area enclosed by the parabola and the line x = 1. This can be done by integrating the parabola equation with respect to x from 0 to 1:
A1 = ∫[0,1] (1 - x)^2 dx
A1 = [x - (x^2)/2 + (x^3)/3] [0,1]
A1 = 1/3
Since A1 - A2 = 1/4, we have:
1/3 - A2 = 1/4
A2 = 1/3 - 1/4
A2 = 1/12
The area A2 is the area enclosed by the parabola and the line x = 1. Therefore, the value of k is such that A2 = 1/12, which corresponds to k = 1/2.
Therefore, the correct answer is option A) 1/2.
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Consider the equations given below:y = (1 - x)2y = 0x = 0A straight line representing x = k separates the area enclosed by the above curves. Say both the areas are A1 (0 ≤ x ≤ k) and A2 (k ≤ x ≤ 1). If A1 - A2 = 1/4, then what is the value of k?a)1/2b)1/4c)2/3d)1/3Correct answer is option 'A'. Can you explain this answer?
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