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In the rectangular coordinate system, what is the area of the region d

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fskilnik
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In the rectangular coordinate system, what is the area of the region d [#permalink] New post   26 Feb 2019, 16:30
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A
B
C
D
E

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85% (hard)

Question Stats:

47% (02:23) correct 53% (02:44) wrong based on 97 sessions
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GMATH practice exercise (Quant Class 20)

\(\left\{ \matrix{\\
\,xy > 0\, \hfill \cr \\
\,0 < y < 2\,\,\,\, \hfill \cr \\
\,x < y + 1 \hfill \cr} \right.\)

In the rectangular coordinate system, what is the area of the region defined by the points (x,y) that satisfy the system of inequalities above?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 5.5
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C
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Re: In the rectangular coordinate system, what is the area of the region d [#permalink] New post   26 Feb 2019, 22:06
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fskilnik wrote:
GMATH practice exercise (Quant Class 20)

\(\left\{ \matrix{\\
\,xy > 0\, \hfill \cr \\
\,0 < y < 2\,\,\,\, \hfill \cr \\
\,x < y + 1 \hfill \cr} \right.\)

In the rectangular coordinate system, what is the area of the region defined by the points (x,y) that satisfy the system of inequalities above?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 5.5



Suggestion : Draw the figure

As we can see that the figure drawn is a trapezium with parallel sides of length 1 and 3 unit lengths and distance between the parallel sides is 2 units

hence area of the shaded region = (1/2)(sum of parallel sides*Distance between them = (1/2)*(1+3)*2 = 4 units

Answer: Option C
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Re: In the rectangular coordinate system, what is the area of the region d [#permalink] New post   27 Feb 2019, 08:25
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fskilnik wrote:
GMATH practice exercise (Quant Class 20)

\(\left\{ \matrix{\\
\,xy > 0\, \hfill \cr \\
\,0 < y < 2\,\,\,\, \hfill \cr \\
\,x < y + 1 \hfill \cr} \right.\)

In the rectangular coordinate system, what is the area of the region defined by the points (x,y) that satisfy the system of inequalities above?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 5.5





\(? = {S_{{\rm{shaded}}}}\,\,\,:\,\,\,\,\left\{ \matrix{\\
\,xy > 0\, \hfill \cr \\
\,0 < y < 2\,\,\,\, \hfill \cr \\
\,x < y + 1 \hfill \cr} \right.\,\,\,\,\,\, \cong \,\,\,\,\,\,\,\,\,\left\{ \matrix{\\
\,x > 0\, \hfill \cr \\
\,0 < y < 2\,\,\,\, \hfill \cr \\
\,y > x - 1 \hfill \cr} \right.\,\)


\(? = {S_{\rm{I}}} + {S_{{\rm{II}}}} = 1 \cdot 2 + {{\left( {3 - 1} \right) \cdot 2} \over 2} = 4\,\,\,\,\, \Rightarrow \,\,\,\,\left( {\rm{C}} \right)\)


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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In the rectangular coordinate system, what is the area of the region d [#permalink] New post   10 Apr 2021, 22:29
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Good question.


The tricky part was analyzing XY> 0

The region covered by this equation is the entire quadrant I and quadrant III where X and Y have the same sign.


0 < Y < 2

The line given by Y = 0 is actually the Horizontal X Axis

The line Give by Y = 2 will be parallel to the X axis and be the Horizontal Line that passes through Y Coordinate of 2 on the Y axis

The region encompassed will be between these 2 parallel lines.


Lastly, X < Y + 1

Rearranging;

Y > X - 1

To analyze this region, first graph Y = X - 1

Y = X will be the line with a 45 degree positive slope that passes through the Origin


Adding (-1) outside the f(x) will shift the Graph of the Line Y = X ——-> 1 unit down on the Y Axis - passes through (0 , -1) Y Intercept.

The region given by Y > X - 1 will be the region to the Upper Left above the line.

The Y axis
The X axis
0 < Y < 2
And
Y > X - 1

All intersect to create a trapezoid in Quadrant 1

The height will be 2 and the 2 parallel sides will be 3 units (along line Y = 2) and 1 unit (along Y = 0)

Area of Trapezoid = (1/2) * (1 + 3) * 2 =

4

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In the rectangular coordinate system, what is the area of the region d [#permalink]
10 Apr 2021, 22:29
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