Which point justifies the shaded area by satisfying $\frac{x^2}{16}+\frac{y^2}{4} \leq 1$?
(1,1)
(4,1)
(-1,2)
(2,-2)
Answer (1)
Substitute each point into the inequality.
Check if the inequality holds true.
( 1 , 1 ) : 16 1 2 + 4 1 2 = 16 5 ≤ 1 (True)
( 4 , 1 ) : 1"> 16 4 2 + 4 1 2 = 4 5 > 1 (False)
( − 1 , 2 ) : 1"> 16 ( − 1 ) 2 + 4 2 2 = 16 17 > 1 (False)
( 2 , − 2 ) : 1"> 16 2 2 + 4 ( − 2 ) 2 = 4 5 > 1 (False)
The point that satisfies the inequality is ( 1 , 1 ) .
Explanation
Understanding the Problem We are given the inequality 16 x 2 + 4 y 2 ≤ 1 and four points: ( 1 , 1 ) , ( 4 , 1 ) , ( − 1 , 2 ) , and ( 2 , − 2 ) . We need to determine which of these points satisfy the inequality.
Checking (1,1) For the point ( 1 , 1 ) , we substitute x = 1 and y = 1 into the inequality: 16 1 2 + 4 1 2 = 16 1 + 4 1 = 16 1 + 16 4 = 16 5 = 0.3125 Since 0.3125 ≤ 1 , the point ( 1 , 1 ) satisfies the inequality.
Checking (4,1) For the point ( 4 , 1 ) , we substitute x = 4 and y = 1 into the inequality: 16 4 2 + 4 1 2 = 16 16 + 4 1 = 1 + 4 1 = 4 5 = 1.25 Since 1"> 1.25 > 1 , the point ( 4 , 1 ) does not satisfy the inequality.
Checking (-1,2) For the point ( − 1 , 2 ) , we substitute x = − 1 and y = 2 into the inequality: 16 ( − 1 ) 2 + 4 2 2 = 16 1 + 4 4 = 16 1 + 1 = 16 17 = 1.0625 Since 1"> 1.0625 > 1 , the point ( − 1 , 2 ) does not satisfy the inequality.
Checking (2,-2) For the point ( 2 , − 2 ) , we substitute x = 2 and y = − 2 into the inequality: 16 2 2 + 4 ( − 2 ) 2 = 16 4 + 4 4 = 4 1 + 1 = 4 5 = 1.25 Since 1"> 1.25 > 1 , the point ( 2 , − 2 ) does not satisfy the inequality.
Conclusion Therefore, only the point ( 1 , 1 ) satisfies the inequality 16 x 2 + 4 y 2 ≤ 1 .
Examples
Understanding inequalities like this is crucial in various fields. For instance, in engineering, it helps define the safe operating area of a machine. Imagine designing a bridge; the load and stress must fall within a certain inequality to ensure the bridge's stability and prevent collapse. Similarly, in economics, budget constraints can be expressed as inequalities, showing the feasible consumption levels given income and prices. This problem demonstrates a fundamental concept with wide-ranging applications.
