Trigonometry Land Plot Calculations
Hey guys! Let's dive into a fascinating trigonometry problem involving a plot of land. We'll break down the calculations step-by-step, making sure everyone understands how to tackle these kinds of questions. We've got a plot of land ABCD with some interesting dimensions and angles, and we're going to figure out the distance from B to D and the fencing requirements. So, grab your calculators, and let's get started!
Problem Overview
Before we jump into the calculations, let’s quickly recap the problem. We have a plot of land, ABCD, with the following properties:
- BC = CD = 60 meters
- ∠BCD = 120°
- ∠ABC = 75°
- ∠ADC = 85°
The problem has two main parts:
- Calculate the distance from B to D: This involves using trigonometric principles to find the length of the diagonal BD across the plot of land. We will need to use the cosine rule and understand how to apply it in different triangles within the plot.
- Determine the fencing requirements: Once we know all the side lengths of the plot, we can calculate the perimeter. Then, knowing the spacing between the poles (3 meters), we can determine the number of poles needed to fence the entire plot. This part involves basic arithmetic and a clear understanding of how to apply the perimeter in a real-world context.
Part 1: Calculating the Distance from B to D
To find the distance from B to D, we'll use the Law of Cosines. This is a super handy tool in trigonometry for finding the sides of a triangle when you know two sides and the included angle. In triangle BCD, we know BC = CD = 60 meters, and ∠BCD = 120°. The Law of Cosines formula is:
BD² = BC² + CD² - 2 * BC * CD * cos(∠BCD)
Let's plug in the values:
BD² = 60² + 60² - 2 * 60 * 60 * cos(120°)
Now, we know that cos(120°) = -0.5. So, the equation becomes:
BD² = 3600 + 3600 - 7200 * (-0.5)
BD² = 7200 + 3600
BD² = 10800
To find BD, we take the square root of 10800:
BD = √10800 ≈ 103.92 meters
So, the distance from B to D is approximately 103.92 meters. This step is crucial because BD acts as a common side for both triangles BCD and ABD. Now that we know BD, we can move on to the next triangle.
Next, we need to find the angles ∠DBC and ∠BDC in triangle BCD. Since BC = CD, triangle BCD is an isosceles triangle. Therefore, ∠DBC = ∠BDC. The sum of angles in a triangle is 180°, so:
∠DBC + ∠BDC + ∠BCD = 180°
2 * ∠DBC + 120° = 180°
2 * ∠DBC = 60°
∠DBC = ∠BDC = 30°
Now we know ∠DBC and ∠BDC, which will be helpful in later calculations. Remember, keeping track of these intermediate results is key to solving the whole problem.
To tackle triangle ABD, we need to find angles ∠ABD and ∠ADB. We know ∠ABC = 75° and ∠ADC = 85°. We've already found ∠DBC = 30°, so:
∠ABD = ∠ABC - ∠DBC
∠ABD = 75° - 30° = 45°
To find ∠ADB, we first need to find ∠BDC, which we already calculated as 30°. Now we can find ∠ADB:
∠ADB = ∠ADC - ∠BDC
However, there seems to be a mistake here. We calculated ∠BDC as 30° within triangle BCD. We need to find ∠ADB by using the angles within quadrilateral ABCD. The sum of angles in a quadrilateral is 360°:
∠ABC + ∠BCD + ∠ADC + ∠BAD = 360°
We know ∠ABC = 75°, ∠BCD = 120°, and ∠ADC = 85°. Plugging these in:
75° + 120° + 85° + ∠BAD = 360°
280° + ∠BAD = 360°
∠BAD = 80°
Now we can use the Law of Cosines in triangle ABD to find the length of AD and AB. We know BD ≈ 103.92 meters and ∠BAD = 80°. Let's rearrange the Law of Cosines formula to solve for AD and AB:
BD² = AB² + AD² - 2 * AB * AD * cos(∠BAD)
This formula is a bit tricky because we have two unknowns (AB and AD). We need additional information or another equation to solve this. Let's try using the Law of Sines in triangle ABD:
AB / sin(∠ADB) = AD / sin(∠ABD) = BD / sin(∠BAD)
We know BD ≈ 103.92 meters and ∠BAD = 80°, and we found ∠ABD = 45°. We still need to find ∠ADB. In triangle ABD, the sum of angles is 180°:
∠ABD + ∠ADB + ∠BAD = 180°
45° + ∠ADB + 80° = 180°
∠ADB = 55°
Now we can use the Law of Sines:
AB / sin(55°) = 103.92 / sin(80°)
AB = (103.92 * sin(55°)) / sin(80°)
AB ≈ (103.92 * 0.819) / 0.985
AB ≈ 86.78 meters
And:
AD / sin(45°) = 103.92 / sin(80°)
AD = (103.92 * sin(45°)) / sin(80°)
AD ≈ (103.92 * 0.707) / 0.985
AD ≈ 74.62 meters
Part 2: Fencing the Plot
Now that we have all the side lengths, we can calculate the perimeter of the plot of land ABCD. The perimeter is the sum of all the sides:
Perimeter = AB + BC + CD + DA
We have:
- AB ≈ 86.78 meters
- BC = 60 meters
- CD = 60 meters
- DA ≈ 74.62 meters
So, the perimeter is:
Perimeter ≈ 86.78 + 60 + 60 + 74.62 ≈ 281.4 meters
The poles are to be placed 3 meters apart. To find the number of poles needed, we divide the perimeter by the distance between the poles:
Number of poles = Perimeter / Distance between poles
Number of poles ≈ 281.4 / 3 ≈ 93.8
Since we can't have a fraction of a pole, we need to round up to the nearest whole number. So, we need 94 poles to fence the plot.
Conclusion
Alright, guys! We've successfully navigated this trigonometry problem. We found that the distance from B to D is approximately 103.92 meters. Then, after calculating all the side lengths, we determined that 94 poles are needed to fence the plot. This problem demonstrates how trigonometry can be applied in real-world scenarios, such as land surveying and construction. Remember, breaking down complex problems into smaller, manageable steps is the key to success. Keep practicing, and you'll become trigonometry pros in no time!
If you found this helpful, give it a thumbs up and let me know what other math problems you'd like to tackle next! Keep learning and keep shining!
