a farmer wishes to fence a field in the form of a right triangle.If one angle of the right triangle is 43.2 degree and the hypotenuse is 200yard,find the amount of fencing needed.
since the hypotenuse is 200, the legs are
200 sin 43.2°
and
200 cos 43.2°
calculate those and just add up the 3 sides.
To find the amount of fencing needed, we need to determine the lengths of the two sides of the right triangle.
Let's assume that the two legs of the right triangle are a and b, with a being the shorter leg and b being the longer leg.
We can use the trigonometric ratios of a right triangle to find the lengths of the legs.
The sine function relates the lengths of the sides to the angles of a right triangle. In this case, we can use the sine function as follows:
sin(43.2 degrees) = opposite/hypotenuse
sin(43.2 degrees) = a/200
To solve for a, we can rearrange the equation:
a = sin(43.2 degrees) * 200
Using a scientific calculator, we find that sin(43.2 degrees) is approximately 0.6830.
So, a ≈ 0.6830 * 200 = 136.6 yards for the shorter leg.
Now, to find the length of the longer leg, we can use the Pythagorean theorem, which states that the sum of the squares of the two legs is equal to the square of the hypotenuse.
a^2 + b^2 = c^2
where a and b are the legs and c is the hypotenuse (given as 200 yards).
Substituting the values we know:
136.6^2 + b^2 = 200^2
Simplifying:
18669.56 + b^2 = 40000
Subtracting 18669.56 from both sides:
b^2 = 21330.44
Taking the square root of both sides to solve for b:
b ≈ √(21330.44)
b ≈ 146.05
So, the length of the longer leg is approximately 146.05 yards.
To find the amount of fencing needed, we add the lengths of the three sides of the right triangle:
fencing needed = a + b + hypotenuse
= 136.6 + 146.05 + 200
= 482.65 yards
Therefore, the amount of fencing needed to enclose the field in the form of a right triangle is approximately 482.65 yards.
To find the amount of fencing needed, we need to find the lengths of the two legs of the right triangle.
Given:
- One angle of the right triangle is 43.2 degrees.
- The hypotenuse is 200 yards.
To determine the lengths of the legs, we can use trigonometric functions.
Step 1: Identify the known and unknown sides of the right triangle.
In a right triangle, the side opposite the given angle is the adjacent side, and the side opposite to the right angle is the hypotenuse. The remaining side is the opposite side.
Known:
- Hypotenuse: 200 yards
- Angle: 43.2 degrees
Unknown:
- Length of the adjacent side (one leg)
- Length of the opposite side (other leg)
Step 2: Use trigonometric functions to find the lengths of the legs.
Since the adjacent side and the hypotenuse are given, we can use the cosine function to find the length of the adjacent side. Similarly, we can use the sine function to find the length of the opposite side.
Cosine function:
cos(angle) = adjacent / hypotenuse
Solve for adjacent:
adjacent = cos(angle) * hypotenuse
Sine function:
sin(angle) = opposite / hypotenuse
Solve for opposite:
opposite = sin(angle) * hypotenuse
Using these formulas, we can find the lengths of the adjacent and opposite sides.
Step 3: Calculate the lengths of the legs.
adjacent = cos(43.2 degrees) * 200 yards
opposite = sin(43.2 degrees) * 200 yards
Using a calculator, we can evaluate these expressions to find:
adjacent ≈ 142.66 yards
opposite ≈ 150.34 yards
Step 4: Find the total amount of fencing needed.
To find the total amount of fencing needed, we need to calculate the perimeter of the triangle, which is the sum of the lengths of the three sides.
Perimeter = adjacent + opposite + hypotenuse
Perimeter = 142.66 yards + 150.34 yards + 200 yards
Using a calculator, we find:
Perimeter ≈ 493 yards
Therefore, the farmer would need approximately 493 yards of fencing.