Question

The second sail has one side of length 22 feet and another of length 2 feet. Determine the range of possible lengths of the third side of the sail.

Answer 1

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Assume x is the longest side of the sail. Use the triangle inequality theorem to write an inequality for the maximum value of the third side and solve for x:

22 + 2 > x

24 > x

Assume x is the shortest side of the sail. Use the triangle inequality theorem to write an inequality for the minimum value of the third side and solve for x:

x + 2 > 22

x > 22 -2

x > 20

Therefore, the range of the possible lengths, x, of the third side of the sail is 20 feet < x < 24 feet.

Explanation:

Answer 2

20 < L < 24

Explanation:

We know that in any given triangle, the length of two sides is always greater than the length of the third side.

Since the sail is a triangle having length of one side as 22 feet and the length of another side as 2 feet, and let L be the length of the third side.

It follows from our triangle rule of sides above that

22 + 2 > L (1)

22 + L > 2 (2)and

L + 2 > 22 (3)

It follows that from (1)

22 + 2 > L

⇒ 24 > L (4)

It follows that from (2)

22 + L > 2

⇒ L > 2 - 22

⇒ L > - 44 (5) and

It follows that from (3)

L + 2 > 22

⇒ L > 22 - 2

⇒ L > 20 (6)

Since from (5) and (6),

L > -44 and L > 20

and 20 > -44 ⇒ L > 20

⇒ 20 < L (7)

From (4) 24 > L ⇒ L < 24 (8)

Combining (7) and (8), we have

20 < L < 24

So, the possible range of values of the third side are 20 < L < 24