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09 Sep 2024, 13:02
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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In the xy-plane, triangular region R is bounded by the lines x = 0, y = 0, and 4x + 3y = 60. Which of the following points lie inside region R ?
Indicate all such points.
A. (2, 18)
B. (5, 12)
C. (10, 7)
D. (12, 3)
E. (15, 2)
Indicate all such points.
A. (2, 18)
B. (5, 12)
C. (10, 7)
D. (12, 3)
E. (15, 2)
08 Feb 2025, 10:54
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OE
Consider the three lines that bound triangular region R. Te line x = 0 is the y-axis, and the line y = 0 is the x-axis. Te line 4x + 3y = 60 intersects the x-axis at (15, 0) and intersects the y-axis at (0, 20). Te fgure below shows region R.
From the fgure, you can see that all points inside region R have positive coordinates and lie below the line 4x + 3y = 60. Note that the equation 4x + 3y = 60 can be rewritten in the form \(y = 20 - \frac{4}{3}x\). In this form, you can see that points inside region R satisfy the inequality \(y < 20 - \frac{4}{3}x\). Since all of the answer choices have positive coordinates, you need only to check whether the coordinates in each answer choice satisfy the inequality \(y < 20 - \frac{4}{3}x\), or equivalently 4x + 3y < 60.
Choice A, (2, 18): 4x + 3y = 4(2) + 3(18) = 62 > 60. So Choice A is not in region R.
Choice B, (5, 12): 4x + 3y = 4(5) + 3(12) = 56 < 60. So Choice B is in region R.
Choice C, (10, 7): 4x + 3y = 4(10) + 3(7) = 61 > 60. So Choice C is in region R.
Choice D, (12, 3): 4x + 3y = 4(12) + 3(3) = 57 < 60. So Choice D is in region R.
Choice E, (15, 2): 4x + 3y = 4(15) + 3(2) = 66 > 60. So Choice E is not in region R.
Thus the correct answer consists of Choices B and D.
1.jpg [ 17.72 KiB | Viewed 164 times ]
Consider the three lines that bound triangular region R. Te line x = 0 is the y-axis, and the line y = 0 is the x-axis. Te line 4x + 3y = 60 intersects the x-axis at (15, 0) and intersects the y-axis at (0, 20). Te fgure below shows region R.
From the fgure, you can see that all points inside region R have positive coordinates and lie below the line 4x + 3y = 60. Note that the equation 4x + 3y = 60 can be rewritten in the form \(y = 20 - \frac{4}{3}x\). In this form, you can see that points inside region R satisfy the inequality \(y < 20 - \frac{4}{3}x\). Since all of the answer choices have positive coordinates, you need only to check whether the coordinates in each answer choice satisfy the inequality \(y < 20 - \frac{4}{3}x\), or equivalently 4x + 3y < 60.
Choice A, (2, 18): 4x + 3y = 4(2) + 3(18) = 62 > 60. So Choice A is not in region R.
Choice B, (5, 12): 4x + 3y = 4(5) + 3(12) = 56 < 60. So Choice B is in region R.
Choice C, (10, 7): 4x + 3y = 4(10) + 3(7) = 61 > 60. So Choice C is in region R.
Choice D, (12, 3): 4x + 3y = 4(12) + 3(3) = 57 < 60. So Choice D is in region R.
Choice E, (15, 2): 4x + 3y = 4(15) + 3(2) = 66 > 60. So Choice E is not in region R.
Thus the correct answer consists of Choices B and D.
Attachment:
1.jpg [ 17.72 KiB | Viewed 164 times ]
