b Start by finding the measure of the angle opposite the 54-foot side.
C
c Recall that the area of any triangle is given by one-half the product of the lengths of two sides times the sine of their included angle.
A
a
B
b The length of the third side of the garden is about 70.4 feet. The angle opposite this side is about 74.4^(∘). The angle opposite the 54-foot side is about 47.6^(∘).
C
c 9 bags
Practice makes perfect
a You are fertilizing a triangular garden. We are asked to draw a diagram representing the situation. One side of the garden is 62 feet long, and other side is 54 feet long. Additionally, the angle opposite the 62-foot side is 58^(∘).
b We are asked to solve the triangle from Part A using the Law of Sines.
Law of Sines
For any triangle, the ratio between the length of the side and the sine value of its opposite angle is constant.
Before we use the law, let's label the vertices of the triangle. We will use the diagram from Part A.
Now we will apply the Law of Sines to the triangle.
a/sinA=62/sin58^(∘)=54/sinCThe above expression can be split into two equations.
a/sinA=62/sin58^(∘) and 62/sin58^(∘)=54/sinC
The first equation involves two unknown values, a and sinA, and the second equation involves one unknown value, sinC. Right now we cannot solve the first equation, so let's start by solving the second equation.
To find the measure of ∠ C, let's use the inverse sine ratio.
sinC=54sin58^(∘)/62
⇕
m∠ C=sin^(- 1)(54sin58^(∘)/62)
We will use a calculator to approximate m∠ C to the nearest tenth of a degree.
m∠ C=sin^(- 1)(54sin58^(∘)/62)≈ 47.6^(∘)
We have that the measure of ∠ C is about 47.6^(∘).
To find the measure of the remaining angle ∠ A, we will use the Triangle Sum Theorem.
m∠ A≈ 180^(∘)- 58^(∘)- 47.6^(∘)=74.4^(∘)
The measure of ∠ A is about 74.4^(∘). We will substitute 74.4^(∘) for A in the first equation and solve it for a.
a/sin 74.4^(∘)≈62/sin58^(∘) ⇔ a≈62sin74.4^(∘)/sin58^(∘)
Finally, let's use a calculator.
a≈62sin74.4^(∘)/sin58^(∘)≈ 70.4
The length of the third side of the triangle is about 70.4 feet.
c To determine how many bags of fertilizer you will need to cover the entire garden, we have to find the area of the garden. Let's consider the following diagram of the garden.
Recall that the area of any triangle is given by one-half the product of the lengths of two sides times the sine of their included angle. In Part B, we found that the included angle of the 62-foot side and the 54-foot side is about 74.4^(∘).
A≈1/2( 62)( 54)sin 74.4^(∘)
Let's estimate the area of the garden.
The area of the garden is about 1612.3 square feet. One bag of fertilizer covers an area of 200 square feet. Let's calculate the number of bags needed to fertilize the entire garden.
1612.3/200=8.0615
Since you cannot purchase a part of a bag, you will need 9 bags of fertilizer for the garden.
