Which graph represents $\frac{(y-3)^2}{25}+\frac{(x+1)^2}{4}>1$ and $y \leq \sqrt{x+5}+2$?
Answer (2)
The problem involves two inequalities: one representing the exterior of an ellipse and the other representing the region below a square root function.
The ellipse is centered at ( − 1 , 3 ) with a vertical major axis of 10 and a horizontal minor axis of 4.
The square root function starts at ( − 5 , 2 ) and opens to the right.
The solution is the intersection of the region outside the ellipse and below the square root function.
Explanation
Analyze the Inequalities We are given two inequalities:
1"> 25 ( y − 3 ) 2 + 4 ( x + 1 ) 2 > 1 , which represents the exterior of an ellipse centered at ( − 1 , 3 ) with a semi-major axis of 5 along the y-axis and a semi-minor axis of 2 along the x-axis.
y ≤ x + 5 + 2 , which represents the region below the graph of the function y = x + 5 + 2 . This is a square root function with a starting point at ( − 5 , 2 ) and opens to the right.
We need to find the graph that represents the intersection of these two regions: outside the ellipse and below the square root function.
Analyze the Ellipse First, let's analyze the ellipse. The center is at ( − 1 , 3 ) . The major axis is vertical with length 2 × 5 = 10 , and the minor axis is horizontal with length 2 × 2 = 4 . The inequality is 'greater than 1', so we are looking for the region outside the ellipse.
Analyze the Square Root Function Next, let's analyze the square root function. The function is y = x + 5 + 2 . The starting point is at x = − 5 , which gives y = − 5 + 5 + 2 = 2 . So the starting point is ( − 5 , 2 ) . The inequality is 'less than or equal to', so we are looking for the region below the curve.
Find the Intersection We need to find the intersection of the region outside the ellipse and the region below the square root function.
Describe the Graph The graph should show an ellipse centered at ( − 1 , 3 ) , with a vertical major axis of length 10 and a horizontal minor axis of length 4. The region outside the ellipse should be shaded. The graph should also show a square root function starting at ( − 5 , 2 ) and opening to the right. The region below the square root function should be shaded. The solution is the intersection of these two shaded regions.
Examples
Understanding inequalities helps in various real-world scenarios. For example, when designing a playground, you might need to ensure that the play area is large enough (outside a certain ellipse) for safety and also below a certain height (square root function) to prevent injuries. The intersection of these conditions defines the safe and usable play area.
The inequalities define regions: the first represents the area outside an ellipse centered at (-1, 3) and the second represents the area below a square root function starting at (-5, 2). The solution is the intersection of these regions, where points satisfy both conditions. A graphical representation will clearly show these shaded areas and their intersections.
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