Circular railway

The railway is to interconnect in a circular arc the points A, B, and C, whose distances are | AB | = 30 km, AC = 95 km, BC | = 70 km. How long will the track from A to C?

Correct result:

b1 =  103.11 km

Solution:

$c=30 \ \\ b=95 \ \\ a=70 \ \\ \ \\ s=\dfrac{ a+b+c }{ 2 }=\dfrac{ 70+95+30 }{ 2 }=\dfrac{ 195 }{ 2 }=97.5 \ \\ S=\sqrt{ s \cdot \ (s-a) \cdot \ (s-b) \cdot \ (s-c) }=\sqrt{ 97.5 \cdot \ (97.5-70) \cdot \ (97.5-95) \cdot \ (97.5-30) } \doteq 672.6522 \ \text{km}^2 \ \\ \ \\ r=S/s=672.6522/97.5 \doteq 6.899 \ \text{km} \ \\ \ \\ R=\dfrac{ a \cdot \ b \cdot \ c }{ 4 \cdot \ r \cdot \ s }=\dfrac{ 70 \cdot \ 95 \cdot \ 30 }{ 4 \cdot \ 6.899 \cdot \ 97.5 } \doteq 74.1468 \ \text{km} \ \\ \ \\ \Delta R-R-b \ \\ b^2=R^2+R^2 - 2 \ R ^2 \ \cos θ \ \\ \ \\ θ=\arccos(\dfrac{ 2 \cdot \ R^2 - b^2 }{ 2 \cdot \ R^2 } )=\arccos(\dfrac{ 2 \cdot \ 74.1468^2 - 95^2 }{ 2 \cdot \ 74.1468^2 } ) \doteq 1.3906 \ \text{rad} \ \\ \ \\ b_{1}=θ \cdot \ R=1.3906 \cdot \ 74.1468=103.11 \ \text{km}$

Try calculation via our triangle calculator.

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 0 comments:
Be the first to comment!

Tips to related online calculators
Cosine rule uses trigonometric SAS triangle calculator.
See also our trigonometric triangle calculator.
Try conversion angle units angle degrees, minutes, seconds, radians, grads.

You need to know the following knowledge to solve this word math problem:

We encourage you to watch this tutorial video on this math problem:

Next similar math problems:

• Square side
If we enlarge the square side a = 5m, its area will increase by 10,25%. How many percent will the side of the square increase? How many percent will it increase the circumference of the square?
• In the
In the rectangle ABCD, the distance of its center from the line AB is 3 cm greater than from the line BC. The circumference of the rectangle is 52 cm. Calculate the contents of the rectangle. Express the result in cm2.
• The bases
The bases of the isosceles trapezoid ABCD have lengths of 10 cm and 6 cm. Its arms form an angle α = 50˚ with a longer base. Calculate the circumference and content of the ABCD trapezoid.
• The circumference
The circumference and width of the rectangle are in a ratio of 5: 1. its area is 216cm2. What is its length?
• Squares ratio
The first square has a side length of a = 6 cm. The second square has a circumference of 6 dm. Calculate the proportions of the perimeters and the proportions of the contents of these squares? (Write the ratio in the basic form). (Perimeter = 4 * a, conte
• Two patches
Peter taped the wound with two rectangular patches (one over the other to form the letter X). The area sealed with both patches at the same time had a content of 40cm2 and a circumference of 30cm. One of the patches was 8cm wide. What was the width of t
• Rectangle field
The field has a shape of a rectangle having a length of 119 m and a width of 19 m. , How many meters have to shorten its length and increase its width to maintain its area and circumference increased by 24 m?
• Triangular prism - regular
The regular triangular prism is 7 cm high. Its base is an equilateral triangle whose height is 3 cm. Calculate the surface and volume of this prism.
• Perimeter of the circle
Calculate the perimeter of the circle in dm, whose radius equals the side of the square containing 0.49 dm2?
• The playground
The playground has the shape of a square with a side of 64 m. It is fenced on three sides. What is the area of the playground and how long is its fence?
• Horses playground
The fence for the horses has the shape of a rectangular trapezoid with an area of 400 m2, the base lengths should be 31 m and 19 m. How many meters of boards will they need to fence it if the boards are stacked in 5 rows?
• Quadrilateral prism
Calculate the surface of a quadrilateral prism according to the input: Area of the diamond base S1 = 2.8 m2, length of the base edge a = 14 dm, height of the prism 1,500 mm.
• The hollow cylinder
The hollow cylinder has a height of 70 cm, an outer diameter of 180 cm and an inner diameter of 120 cm. What is the surface of the body, including the area inside the cavity?
• Carpet
How many crowns CZK do we pay for a carpet for a bedroom, when 1m of square carpet costs 350 CZK and the bedroom has dimensions of 4m and 6m? How many crowns do we pay for a strip around the carpet, when 1m of the strip costs 15 CZK?
• Triangular prism
The base of the perpendicular triangular prism is a rectangular triangle with a hypotenuse of 10 cm and one leg of 8 cm. The prism height is 75% of the perimeter of the base. Calculate the volume and surface of the prism.
• Cross five
The figure on the picture is composed of the same squares and has a content of 45cm². What's its perimeter?
• Rectangular garden
The perimeter of Peter's rectangular garden is 98 meters. The width of the garden is 60% shorter than its length. Find the dimensions of the rectangular garden in meters. Find the garden area in square meters.