Question

2b) A farmer is building a pen inside a barn. The pen will be in the shape of a right triangle.

The farmer has 14 feet of barn wall to use for one side of the pen and wants another side of the pen to be 15 feet long.


a. To the nearest tenth of a foot, find all possible lengths for the third side of the triangle.


b. The farmer wants the area of the pen to be as large as possible. What length should he choose for the third side? Justify your answer.

Answer 1

a. c ≈ 20.5 ft (first possibility)

b ≈ 5.4 ft(other possibility)

b. The farmer should choose the side that has 20.5 ft for the third side because it provide more larger area as needed by the farmer.

Explanation:

The pen the farmer wants to build is a right angle triangle. one side of the triangle is 14 ft while another side is 15 ft.

A right angle triangle has opposite side, adjacent side and an hypotenuse which is the longest side.

a. To the nearest tenth of a foot, find all possible lengths for the third side of the triangle.

Pythagoras theorem can be used to solve any sides of the triangle when given 2 sides.

We are not told which side is the hypotenuse or the adjacent or the opposites side. Therefore , the possible length for the third side can be computed using Pythagoras theorem.

c² = a² + b²

c = hypotenuse

while a and b can be any of opposite or adjacent sides. The first possible length can be when both sides are the legs of the triangle (no hypotenuse)

c² = 14² + 15²

c² = 196 + 225

c² = 421

c= √421

c = 20.5182845287

c ≈ 20.5 ft (first possibility)

The other possibility of the third side is when the hypotenuse is Known and one other side(either adjacent or opposite). We can use only 15 since it should be the longest side.

c² = a² + b²

c² - a² = b²

15² - 14² = b²

225 - 196 = b²

b² = 29

b = √29

b = 5.38516480713

b ≈ 5.4 ft(other possibility)

b. The farmer wants the area of the pen to be as large as possible. What length should he choose for the third side? Justify your answer.

The two scenarios are as follows

First area

area = 1/2 × base × height

area = 1/2 × 14 × 15 = 210/2 = 105 ft²

Second area

area = 1/2 × base × height

area = 1/2 × 5.4 × 14 = 75.6/2 = 37.8 ft²

The farmer should choose the side that has 20.5 ft for the third side because it provide more larger area as needed by the farmer.

Answer 2

For the pen to be as large as possible, the length the farmer should choose for the third side should be 20.52 feet

Explanation:

The length of one side of the right triangle = 14 feet

The length of the other side = 15 feet

Therefore, the length of the third side can be one of the following side lengths;

`S_(3,1) = √(14^2 + 15^2) = √(421) = 20.52 \ feet`

`S_(3,2) = √(15^2 - 14^2) = √(29) = 5.39 \ feet`

The possible lengths of the third side of the triangle are;

Third side = 20.52 feet and

Third side = 5.39 feet

b. For the area of the pen to be as large as possible, we have;

With third side = 20.52 feet, area of the pen = 0.5 × 15 × 14 = 105 ft²

With third side = 5.39 feet, area of the pen = 0.5 × 5.39 × 14 = 37.7 ft²

Therefore, for the pen to be as large as possible, the farmer should choose 20.52 feet as he third side.