A landscaper is building a triangular garden behind a house. The lot is bounded on one side by the house that is 30 ft. long and on another side by a fence that is 50 ft. long. If the fence makes a 25 degree angle with the house, how long is the third side of the garden? Round your answer to the nearest foot.

Question

A landscaper is building a triangular garden behind a house. The lot is bounded on one side by the house that is 30 ft. long and on another side by a fence that is 50 ft. long. If the fence makes a 25 degree angle with the house, how long is the third side of the garden? Round your answer to the nearest foot.

Part 1) Which law is needed to find the missing side?

A) Law of Sines
B) Law of Cosines

Part 2) Show all work to find the missing side. Make sure to put what equation you are using and the work to solve it.

Answer 1

Part 1) We need to use the Law of Cosines to find the missing side.

Part 2) Let's label the missing side as c. According to the Law of Cosines:

c^2 = a^2 + b^2 - 2ab cos(C)

Where a and b are the known sides adjacent to the missing side, and C is the angle opposite to the missing side. In this case, a = 30 ft, b = 50 ft, and C = 25 degrees. Substituting these values:

c^2 = 30^2 + 50^2 - 2(30)(50)cos(25)

c^2 = 1062.44

Taking the square root of both sides:

c ≈ 32.6 ft

Therefore, the third side of the garden is approximately 32.6 ft long.

Answer 2

Part 1) To find the missing side of the triangular garden, we need to use the Law of Cosines.

Part 2) The Law of Cosines is used to find the length of a side when we know the lengths of the other two sides and the angle between them.

Let's denote the missing side of the triangular garden as side c.

In this case, side a is the length of the fence that is 50 ft, side b is the length of the house that is 30 ft, and angle A is the angle between sides a and b, which is 25 degrees.

The Law of Cosines is given by:

c^2 = a^2 + b^2 - 2ab * cos(A)

Plugging in the given values in the equation, we get:

c^2 = 50^2 + 30^2 - 2 * 50 * 30 * cos(25)

c^2 = 2500 + 900 - 3000 * cos(25)

Now, we can calculate c by taking the square root of both sides of the equation:

c = √(2500 + 900 - 3000 * cos(25))

Using a calculator, we can evaluate the expression:

c ≈ 46.81 ft

Rounding this to the nearest foot, the length of the third side of the garden is approximately 47 ft.

Answer 3

Part 1) To find the missing side of the garden, we need to use the Law of Cosines.

Part 2) Let's label the sides of the garden as follows:

- Side A: Length of the side of the garden adjacent to the 25-degree angle (Unknown)
- Side B: Length of the side of the garden adjacent to the house (30 ft)
- Side C: Length of the side of the garden adjacent to the fence (50 ft)

The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)

In this case, we are looking for side A, so let's substitute the values we have:
A^2 = 30^2 + 50^2 - 2 * 30 * 50 * cos(25°)

Now, let's calculate the missing side using this equation:

A^2 = 900 + 2500 - 3000 * cos(25°)
A^2 = 3400 - 3000 * cos(25°)
A^2 ≈ 3400 - 3000 * 0.9063 (rounding cos(25°) to four decimal places)
A^2 ≈ 3400 - 2718.90
A^2 ≈ 681.10

Finally, we take the square root of both sides to find the length of side A:

A ≈ √(681.10)
A ≈ 26.11

Therefore, the length of the third side (side A) of the garden is approximately 26 feet.