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Rohan is checking in poodles for a dog show. A miniature poodle must be between 10 in. and 15 in. at the shoulder. The body length must be no more than 16 in. In addition, the poodle’s shoulder height must be no more than 1 in. longer than its body length.

The graph shows the feasible region, where x represents the poodle’s body length and y represents the shoulder height.

Which ordered pairs meet all the constraints for a successful poodle measurement and make sense in context of the situation?

Select each correct answer.

Responses

(11, 9)
begin ordered pair 11 comma 9 end ordered pair

(10, 12)
begin ordered pair 10 comma 12 end ordered pair

(15, 11)
begin ordered pair 15 comma 11 end ordered pair

(12, 10.5)
begin ordered pair 12 comma 10.5 end ordered pair

(11, 14)
begin ordered pair 11 comma 14 end ordered pair

1 answer

To determine which ordered pairs meet all the constraints for a miniature poodle, let's summarize the constraints based on the problem:

  1. The shoulder height \( y \) (in inches) must be between 10 inches and 15 inches: \[ 10 \leq y \leq 15 \]

  2. The body length \( x \) (in inches) must be no more than 16 inches: \[ x \leq 16 \]

  3. The shoulder height must be no more than 1 inch longer than the body length: \[ y \leq x + 1 \]

Now, we will analyze each ordered pair:

  1. (11, 9)

    • \( y = 9 \) does not satisfy \( 10 \leq y \leq 15 \).
    • Not valid.

  2. (10, 12)

    • \( 10 \leq y = 12 \leq 15 \) (valid for height).
    • \( x = 10 \leq 16 \) (valid for body length).
    • \( y = 12 \leq x + 1 = 10 + 1 = 11 \) (not valid for height).
    • Not valid.

  3. (15, 11)

    • \( 10 \leq y = 11 \leq 15 \) (valid for height).
    • \( x = 15 \leq 16 \) (valid for body length).
    • \( y = 11 \) is less than \( x + 1 = 15 + 1 = 16 \) (valid for height and length relation).
    • Valid.

  4. (12, 10.5)

    • \( 10 \leq y = 10.5 \leq 15 \) (valid for height).
    • \( x = 12 \leq 16 \) (valid for body length).
    • \( y = 10.5 \leq x + 1 = 12 + 1 = 13 \) (valid for height and length relation).
    • Valid.

  5. (11, 14)

    • \( 10 \leq y = 14 \leq 15 \) (valid for height).
    • \( x = 11 \leq 16 \) (valid for body length).
    • \( y = 14 \leq x + 1 = 11 + 1 = 12 \) (not valid for height and length relation, as 14 is not less than or equal to 12).
    • Not valid.

Based on the analysis, the valid ordered pairs are:

  • (15, 11)
  • (12, 10.5)

So the final answers are:

  • (15, 11)
  • (12, 10.5)