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12 Jul 2022, 08:00
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Difficulty:
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Question Stats:
34% (02:19) correct
66%
(03:04)
wrong
based on 177
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A green rectangle is inscribed in a right triangle as shown above. What is the value of (the minimum possible area of the triangle) PLUS (the area of the green rectangle)?
A. 60
B. 90
C. 120
D. 180
E. 240
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12 Jul 2022, 09:06
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Given: A green rectangle is inscribed in a right triangle as shown above.
Asked: What is the value of (the minimum possible area of the triangle) PLUS (the area of the green rectangle)?
Since triangles ADF and FEC are similar
AD/DF = FE/EC
12/x = y/5
xy = 60
The area of the triangle + the area of the green rectangle = (12+y)(x+5)/2 + xy = (12 + 60/x)(x+5)/2 + 60 =
(12x + 60 + 60 + 300/x)/2 + 60 = 120 + 6x + 150/x
To minimize (The area of the triangle + the area of the green rectangle), let us find out first derivative of the expression and equate to 0.
6 - 150/x^2 = 0
x^2 = 150/6 = 25
x = 5
To minimize, The second derivative should be positive
300/x^3 > 0 : since x = 5>0
Minimum (The area of the triangle + the area of the green rectangle) = 120 + 6*5 + 150/5 = 120 + 30 + 30 = 180
IMO D
General Discussion
Let us say that length of rectangle is y and breadth is x. Now since the small triangle above the green rectangle and the big one is similar we can say that:
x/x+5 = 12/12+y
When we cross multiply: 12x+xy = 12x + 60
so we can say that xy=60 (eq 1)
Now we have find total area of big triangle and the green rectangle: 1/2*x*12 + 2*x*y + 1/2*5*y
We know the value of 2xy will be 120. So options A, B, and C are eliminated since area is bound to be more than 120.
If we plug in value for option D which is 180 we get:
1/2*x*12 + 2*x*y + 1/2*5*y = 180
1/2*x*12 + 1/2*5*y = 180 - 120
6x + 2.5y = 60
We can substitute the value of y from eq 1:
6x + 2.5 (60/x) = 60
Solving this we will get value of x as 5, and value of y as 12.
To double check when we plus these values in the equation for total area (1/2*x*12 + 2*x*y + 1/2*5*y) we get it as equal to 180. Since we are asked to find the minimum possible area we don't have to check for the last option E.
D is the answer
x/x+5 = 12/12+y
When we cross multiply: 12x+xy = 12x + 60
so we can say that xy=60 (eq 1)
Now we have find total area of big triangle and the green rectangle: 1/2*x*12 + 2*x*y + 1/2*5*y
We know the value of 2xy will be 120. So options A, B, and C are eliminated since area is bound to be more than 120.
If we plug in value for option D which is 180 we get:
1/2*x*12 + 2*x*y + 1/2*5*y = 180
1/2*x*12 + 1/2*5*y = 180 - 120
6x + 2.5y = 60
We can substitute the value of y from eq 1:
6x + 2.5 (60/x) = 60
Solving this we will get value of x as 5, and value of y as 12.
To double check when we plus these values in the equation for total area (1/2*x*12 + 2*x*y + 1/2*5*y) we get it as equal to 180. Since we are asked to find the minimum possible area we don't have to check for the last option E.
D is the answer
