please help

two boundary lines of a piece of property intersect at an angle 85 deg.. it is desired to cut off a triangular portion of the property which will be one acre (43560 sq. ft.) in area by means of a straight fence. If the fence begins at a point on one boundary 250 ft, from the corner of the property , and runs in a straight line to the other boundary , what does it make with the boundary lines and how long is it?

Draw a diagram. Let the 1-acre parcel be triangle ABC, with

B=85°
AB = 250
h is the altitude from AB to C
x is the length of side BC
y is side AC

250h = 43560
h = 174.24
h/x = sin85°
x = 174.91
y^2 = 250^2 + x^2 - 500x cos85°
y=292.36

x/sinθ = y/sin85°
sinθ = 174.91 sin85° / 292.36 = 0.60
θ = 36.6°

So, the fence is 292.36 ft long, making angles of 36.6° and 58.4° with the other sides.

rats - 1/2 * 250h = 43560

so, h=348.8

Adjust the other calculations accordingly. Good exercise for the student :-)

To solve this problem, we can use trigonometry and the concept of similar triangles. Here are the step-by-step instructions:

Step 1: Draw a diagram representing the property. Label the two boundary lines that intersect at an angle of 85 degrees and the point where the fence starts.

Step 2: From the given information, we know that the triangular portion of the property has an area of 43560 square feet (1 acre). This triangle is formed by the two boundary lines and the straight fence.

Step 3: We can calculate the length of the straight fence using the formula for the area of a triangle:

Area of triangle = (1/2) * base * height

In this case, the base of the triangle is the length of the straight fence and the height is the distance from the starting point to the intersection of the two boundary lines.

So, 43560 sq. ft. = (1/2) * length of fence * height

Step 4: We need to find the height of the triangle. To do this, we need to find the length of the line segment from the starting point to the intersection of the two boundary lines. Let's call this length "x."

Step 5: Now, we can use the concept of similar triangles to determine the relationship between the lengths of the two boundary lines and the length of the line segment x.

Since the two boundary lines form an angle of 85 degrees, the line segment x forms an angle of (180 - 85) = 95 degrees with one of the boundary lines.

In a right triangle formed by the line segment x, the height (distance from the starting point to the intersection of the two boundary lines), and the length of one of the boundary lines, the angle between the line segment x and the height is 90 degrees.

Using trigonometry, we can write the equation:

tan(95 degrees) = height / length of one boundary line

From this equation, we can solve for the height in terms of x.

Step 6: Substituting the height in terms of x into the equation from Step 3, we get:

43560 sq. ft. = (1/2) * length of fence * (height in terms of x)

Simplifying this equation will give us the length of the fence.

Step 7: Finally, we can use the trigonometric properties to find the angle that the fence makes with the boundary lines. This angle will be the same as the angle between the line segment x and the boundary line.

Use the equation:

tan(angle) = height / (length of one boundary line - x)

Finally, solve for the angle.

By following these steps, you should be able to determine the angle the fence makes with the boundary lines and the length of the fence.

To find the angle that the fence makes with the boundary lines, we can use basic trigonometry. Let's denote the angle that the fence makes with the first boundary line as α.

First, we need to draw a diagram to visualize the situation. The property forms a triangle, with two boundary lines intersecting at a corner. The fence starts at a point on one boundary line, 250 ft away from the corner, and runs to the other boundary line.

/
Fence \ α
/
/_________
Property

From the information given, we know that the property has an area of one acre, which is equal to 43,560 square feet. The formula for the area of a triangle is 0.5 * base * height. In this case, the base of the triangle is the length of the fence and the height is the perpendicular distance from the starting point of the fence to the opposite boundary line.

Let's denote the length of the fence as x, the perpendicular distance from the starting point of the fence to the opposite boundary line as y, and the distance from the starting point of the fence to the corner of the property as z.

We can set up the following equations based on the given information:

1) The area of the triangle: (0.5) * x * y = 43,560 square feet
2) The cosine rule: y^2 = z^2 + (250 ft)^2 - 2 * z * 250 ft * cos(85 deg)

We have two equations and two unknowns (x and y), so we can solve this system of equations to find the values of x and y.

First, let's solve equation 2) for y:

y^2 = z^2 + (250 ft)^2 - 2 * z * 250 ft * cos(85 deg)
y^2 = z^2 + 250^2 - 2 * 250 * z * cos(85 deg)
y^2 = z^2 + 250^2 - 500 * z * cos(85 deg)

Substituting this expression for y^2 into equation 1), we get:

(0.5) * x * (z^2 + 250^2 - 500 * z * cos(85 deg)) = 43,560 square feet
x * (z^2 + 250^2 - 500 * z * cos(85 deg)) = 2 * 43,560 square feet
x * (z^2 + 62,500 - 500 * z * cos(85 deg)) = 87,120 square feet

Now, we have an equation in terms of x and z. We can substitute the given values and solve for x.

Let's assume that z is given as a specific value, for example, z = 300 ft.

x * (300^2 + 62,500 - 500 * 300 * cos(85 deg)) = 87,120 square feet
x * (90,000 + 62,500 - 150,000 * cos(85 deg)) = 87,120 square feet
x * (152,500 - 150,000 * cos(85 deg)) = 87,120 square feet
x = 87,120 square feet / (152,500 - 150,000 * cos(85 deg))

Using a calculator, we can evaluate this expression to find the value of x.

Once we have found the value of x, we can use trigonometry to find the angle α. We can use the sine rule:

sin(α) = y / x

Substituting the values of y and x, we can solve for α.

Finally, we have the angle that the fence makes with the boundary lines (α) and the length of the fence (x).

Please note that these calculations are based on the assumption of a specific value for z. If z is not given, you may need to use a different approach or gather additional information to solve the problem.