A Farmer Has A Triangular Field Where Two Sides Measure 458 M And 358 M. The Angle Between These Two Sides Measures $80^{\circ}$. The Farmer Wishes To Use An Insecticide That Costs $K 4.58$ Per $ 180 M 2 180 M^2 180 M 2 [/tex] Or

Mar 8, 2025 by ADMIN 234 views

$A farmer has a triangular field where two sides measure 458 m and 358 m. The angle between these two sides measures $80^{\circ}$. The farmer wishes to use an insecticide that costs $K 4.58$ per \$180 m^2$.$

Calculating the Area of the Triangular Field

To determine the cost of the insecticide needed to cover the entire field, we first need to calculate the area of the triangular field. The formula for the area of a triangle is given by:

Area = (1/2) * base * height

However, we are given two sides and the angle between them, not the height. We can use trigonometry to find the height of the triangle.

Using Trigonometry to Find the Height

We can use the sine function to find the height of the triangle. The sine function is defined as:

sin(angle) = opposite side / hypotenuse

In this case, the angle is 80 degrees, the opposite side is the height we want to find, and the hypotenuse is the side opposite the angle, which we can call c.

Finding the Length of the Third Side

To use the sine function, we need to find the length of the third side of the triangle. We can use the law of cosines to do this:

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

where a and b are the two given sides, and c is the length of the third side.

Calculating the Area of the Triangular Field

Now that we have the height of the triangle, we can use the formula for the area of a triangle to find the area of the field.

Determining the Cost of the Insecticide

Once we have the area of the field, we can determine the cost of the insecticide needed to cover the entire field.

Step-by-Step Solution

Step 1: Find the Length of the Third Side

First, we need to find the length of the third side of the triangle using the law of cosines.

c^2 = 458^2 + 358^2 - 2 * 458 * 358 * cos(80)

c^2 = 210244 + 128244 - 2 * 458 * 358 * 0.173648

c^2 = 338488 - 2 * 164004.784

c^2 = 338488 - 328009.568

c^2 = 10478.432

c = sqrt(10478.432)

c = 102.3 m

Step 2: Find the Height of the Triangle

Now that we have the length of the third side, we can use the sine function to find the height of the triangle.

sin(80) = height / 458

height = 458 * sin(80)

height = 458 * 0.9848

height = 451.3 m

Step 3: Calculate the Area of the Triangular Field

Now that we have the height of the triangle, we can use the formula for the area of a triangle to find the area of the field.

Area = (1/2) * base * height

Area = (1/2) * 458 * 451.3

Area = 103, 511.9 m^2

Step 4: Determine the Cost of the Insecticide

Now that we have the area of the field, we can determine the cost of the insecticide needed to cover the entire field.

Cost = (Area / 180) * 4.58

Cost = (103, 511.9 / 180) * 4.58

Cost = 574.3 * 4.58

Cost = 2, 630.3 K

Conclusion

In conclusion, the area of the triangular field is approximately 103, 511.9 m^2, and the cost of the insecticide needed to cover the entire field is approximately 2, 630.3 K.

Discussion

The problem of finding the area of a triangle given two sides and the angle between them is a classic problem in trigonometry. The solution involves using the law of cosines to find the length of the third side, and then using the sine function to find the height of the triangle. The area of the triangle can then be found using the formula for the area of a triangle.

Real-World Applications

This problem has many real-world applications, such as finding the area of a field for agricultural purposes, or finding the area of a roof for construction purposes.

Future Work

In the future, it would be interesting to explore other methods for finding the area of a triangle given two sides and the angle between them. For example, one could use the law of sines to find the area of the triangle.

References

"Trigonometry" by Michael Corral
"Geometry" by Michael Spivak

Code

import math
def find_area(a, b, angle):
# Find the length of the third side using the law of cosines
c = math.sqrt(a2 + b2 - 2ab*math.cos(math.radians(angle)))
# Find the height of the triangle using the sine function
height = a * math.sin(math.radians(angle))

# Calculate the area of the triangle
area = 0.5 * a * height

return area

a = 458
b = 358
angle = 80
area = find_area(a, b, angle)
print("The area of the triangle is:", area)

Conclusion

In conclusion, the area of the triangular field is approximately 103, 511.9 m^2, and the cost of the insecticide needed to cover the entire field is approximately 2, 630.3 K. The problem of finding the area of a triangle given two sides and the angle between them is a classic problem in trigonometry, and has many real-world applications.

Introduction

In our previous article, we discussed how to find the area of a triangle given two sides and the angle between them. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the formula for finding the area of a triangle given two sides and the angle between them?

A: The formula for finding the area of a triangle given two sides and the angle between them is:

Area = (1/2) * a * b * sin(angle)

where a and b are the two given sides, and angle is the angle between them.

Q: How do I find the length of the third side of the triangle?

A: To find the length of the third side of the triangle, you can use the law of cosines:

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

where c is the length of the third side, a and b are the two given sides, and angle is the angle between them.

Q: What if I don't have a calculator? Can I still find the area of the triangle?

A: Yes, you can still find the area of the triangle without a calculator. You can use a trigonometric table or a trigonometric chart to find the sine of the angle. Alternatively, you can use a method called "estimation" to find the area of the triangle.

Q: How do I use estimation to find the area of the triangle?

A: To use estimation to find the area of the triangle, you need to make an educated guess about the height of the triangle. You can do this by drawing a diagram of the triangle and using your knowledge of geometry to estimate the height. Once you have an estimate of the height, you can use the formula for the area of a triangle to find the area.

Q: What if I have a right triangle? Can I still use the formula for the area of a triangle given two sides and the angle between them?

A: Yes, you can still use the formula for the area of a triangle given two sides and the angle between them even if you have a right triangle. In this case, the angle between the two sides is 90 degrees, and the sine of 90 degrees is 1. So, the formula simplifies to:

Area = (1/2) * a * b

Q: Can I use the formula for the area of a triangle given two sides and the angle between them to find the area of a rectangle?

A: No, you cannot use the formula for the area of a triangle given two sides and the angle between them to find the area of a rectangle. The formula is specifically designed for triangles, and it will not give you the correct answer for a rectangle.

Q: What if I have a triangle with two sides of equal length? Can I still use the formula for the area of a triangle given two sides and the angle between them?

A: Yes, you can still use the formula for the area of a triangle given two sides and the angle between them even if you have a triangle with two sides of equal length. In this case, the angle between the two sides is 90 degrees, and the sine of 90 degrees is 1. So, the formula simplifies to:

Area = (1/2) * a^2

Q: Can I use the formula for the area of a triangle given two sides and the angle between them to find the area of a circle?

A: No, you cannot use the formula for the area of a triangle given two sides and the angle between them to find the area of a circle. The formula is specifically designed for triangles, and it will not give you the correct answer for a circle.

Q: What if I have a triangle with a very small angle? Can I still use the formula for the area of a triangle given two sides and the angle between them?

A: Yes, you can still use the formula for the area of a triangle given two sides and the angle between them even if you have a triangle with a very small angle. However, you may need to use a calculator to find the sine of the angle, as the sine of a very small angle is very close to zero.

Q: Can I use the formula for the area of a triangle given two sides and the angle between them to find the area of a polygon with more than three sides?

A: No, you cannot use the formula for the area of a triangle given two sides and the angle between them to find the area of a polygon with more than three sides. The formula is specifically designed for triangles, and it will not give you the correct answer for a polygon with more than three sides.

Q: What if I have a triangle with a very large angle? Can I still use the formula for the area of a triangle given two sides and the angle between them?

A: Yes, you can still use the formula for the area of a triangle given two sides and the angle between them even if you have a triangle with a very large angle. However, you may need to use a calculator to find the sine of the angle, as the sine of a very large angle is very close to 1.

Conclusion

In conclusion, the formula for finding the area of a triangle given two sides and the angle between them is a powerful tool that can be used to solve a wide range of problems. However, it is essential to understand the limitations of the formula and to use it only when it is applicable.

References

"Trigonometry" by Michael Corral
"Geometry" by Michael Spivak

Code

import math
def find_area(a, b, angle):
# Find the length of the third side using the law of cosines
c = math.sqrt(a2 + b2 - 2ab*math.cos(math.radians(angle)))
# Find the height of the triangle using the sine function
height = a * math.sin(math.radians(angle))

# Calculate the area of the triangle
area = 0.5 * a * height

return area


a = 458
b = 358
angle = 80
area = find_area(a, b, angle)
print("The area of the triangle is:", area)

Discussion

The formula for finding the area of a triangle given two sides and the angle between them is a fundamental concept in geometry. It has many real-world applications, such as finding the area of a field for agricultural purposes, or finding the area of a roof for construction purposes. However, it is essential to understand the limitations of the formula and to use it only when it is applicable.

Future Work

References

"Trigonometry" by Michael Corral
"Geometry" by Michael Spivak

Code

import math
def find_area(a, b, angle):
# Find the length of the third side using the law of cosines
c = math.sqrt(a2 + b2 - 2ab*math.cos(math.radians(angle)))
# Find the height of the triangle using the sine function
height = a * math.sin(math.radians(angle))

# Calculate the area of the triangle
area = 0.5 * a * height

return area


a = 458
b = 358
angle = 80
area = find_area(a, b, angle)
print("The area of the triangle is:", area)