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?

To determine the ordered pairs (x, y) that satisfy all the constraints for a successful miniature poodle measurement, let's define the constraints clearly based on the problem.

1. Shoulder Height (y): The shoulder height must be between 10 inches and 15 inches, which gives us the inequality: \[ 10 \leq y \leq 15 \]

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

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

Combining these constraints forms a feasible region in a coordinate graph where the x-axis represents body length and the y-axis represents shoulder height.

To find the ordered pairs that satisfy all these constraints:

1. From the height constraints: The poodle's shoulder height (y) must be at least 10 and no more than 15.

2. From the body length constraint: The maximum length is 16; therefore x can range from any non-negative number (generally 0 onward for practical measurements) to 16.

3. From the relationship of y and x:

   • Since \( y \leq x + 1\), for a specific body length (x), the shoulder height (y) must be at most 1 inch greater than that body length.
   • If the body length is 16, the maximum shoulder height can be \( 16 + 1 = 17\), but since it maxes at 15, we will be capped there.
   • The poodle's shoulder height must stay within the 10 to 15 height range.

4. Analyzing the scenario:

   • Pair (0 ≤ x ≤ 16) with (10 ≤ y ≤ 15) and \(y \leq x + 1\).
   • The effective range for y now becomes \(10 \leq y \leq 15\) AND \(y \leq x + 1\).

Finding Ordered Pairs

• At x = 14:

  • \(y \leq 14 + 1 = 15\) means y can be up to 15. Shifted vertically down to \(y = 15\):
  • Ordered pair: (14, 15)

• At x = 15:

  • \(y \leq 15 + 1 = 16\) but max is 15.
  • Ordered pair: (15, 15)

• At x = 16:

  • \(y \leq 16 + 1 = 17\), capped at 15.
  • Ordered pair: (16, 15).

• Maximizing Values within range:

  • (14, 14)
  • (14, 13)
  • (14, 12)
  • (14, 11)
  • (14, 10)

• No values below for (x ≤ 10) but upwards through the range.

Conclusion:

Feasible ordered pairs that meet all the constraints are:

• (14, 10)
• (14, 11)
• (14, 12)
• (14, 13)
• (14, 14)
• (14, 15)
• (15, 10)
• (15, 11)
• (15, 12)
• (15, 13)
• (15, 14)
• (15, 15)
• (16, 10)
• (16, 11)
• (16, 12)
• (16, 13)
• (16, 14)
• (16, 15)

These pairs reflect potential body lengths and shoulder heights in the defined constraints for the poodle measurements.