A bridge

A bridge over a river is in the shape of the arc of a circle with each base of the bridge at the river's edge. At the center of the river, the bridge is 10 feet above the water. At 27 feet from the edge of the river, the bridge is 9 feet above the water. How wide is the river?

Result

a =  80 ft

Solution:

r=10+x r2=x2+(27+y)2 r2=(9+x)2+y2  (10+x)2=x2+(27+y)2 (10+x)2=(9+x)2+y2  20 x=y2+54 y+629 2 x+19=y2   20⋅ (y2−19)/2=y2+54 y+629 9y2−54y−819=0  a=9;b=−54;c=−819 D=b2−4ac=542−4⋅9⋅(−819)=32400 D>0  y1,2=−b±D2a=54±3240018 y1,2=54±18018 y1,2=3±10 y1=13 y2=−7   Factored form of the equation:  9(y−13)(y+7)=0  x>0;y>0;r>0  y=y1=13 ft x=(y2−19)/2=(132−19)/2=75 ft r=10+x=10+75=85 ft  a=2⋅ 27+2⋅ y=2⋅ 27+2⋅ 13=80 ftr=10+x \ \\ r^2=x^2+(27+y)^2 \ \\ r^2=(9+x)^2 + y^2 \ \\ \ \\ (10+x)^2=x^2+(27+y)^2 \ \\ (10+x)^2=(9+x)^2 + y^2 \ \\ \ \\ 20 \ x=y^2 + 54 \ y + 629 \ \\ 2 \ x + 19=y^2 \ \\ \ \\ \ \\ 20 \cdot \ (y^2-19)/2=y^2 + 54 \ y + 629 \ \\ 9y^2 -54y -819=0 \ \\ \ \\ a=9; b=-54; c=-819 \ \\ D=b^2 - 4ac=54^2 - 4\cdot 9 \cdot (-819)=32400 \ \\ D>0 \ \\ \ \\ y_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 54 \pm \sqrt{ 32400 } }{ 18 } \ \\ y_{1,2}=\dfrac{ 54 \pm 180 }{ 18 } \ \\ y_{1,2}=3 \pm 10 \ \\ y_{1}=13 \ \\ y_{2}=-7 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 9 (y -13) (y +7)=0 \ \\ \ \\ x >0;y>0;r>0 \ \\ \ \\ y=y_{1}=13 \ \text{ft} \ \\ x=(y^2-19)/2=(13^2-19)/2=75 \ \text{ft} \ \\ r=10+x=10+75=85 \ \text{ft} \ \\ \ \\ a=2 \cdot \ 27 + 2 \cdot \ y=2 \cdot \ 27 + 2 \cdot \ 13=80 \ \text{ft}

Checkout calculation with our calculator of quadratic equations.


Try another example


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 2 comments:
#
Dr Math
THE BRIDGE OVER THE RIVER

#
Dr Math
r = radius of circle of arc

x = distance between center of circle and water level (center of the circle is under ground)

y = distance in horiznotal of point on circle which is 9 ft above the water

avatar









Tips to related online calculators
Looking for help with calculating roots of a quadratic equation?
Do you have a system of equations and looking for calculator system of linear equations?
Pythagorean theorem is the base for the right triangle calculator.
See also our trigonometric triangle calculator.

Following knowledge from mathematics are needed to solve this word math problem:

Next similar math problems:

  1. Pavement
    chodnik2 Calculate the length of the pavement that runs through a circular square with a diameter of 40 m if distance the pavement from the center is 15 m.
  2. RT and circles
    r_triangle Solve right triangle if the radius of inscribed circle is r=9 and radius of circumscribed circle is R=23.
  3. Guppies for sale
    guppies Paul had a bowl of guppies for sale. Four customers were milling around the store. 1. Rod told paul - I'll take half the guppies in the bowl, plus had a guppy. 2. Heather said - I'll take half of what you have, plus half a guppy. The third customer, Na
  4. Linsys2
    linear_eq_3 Solve two equations with two unknowns: 400x+120y=147.2 350x+200y=144
  5. Three unknowns
    matrix_1 Solve the system of linear equations with three unknowns: A + B + C = 14 B - A - C = 4 2A - B + C = 0
  6. Expression with powers
    eq222_9 If x-1/x=5, find the value of x4+1/x4
  7. Three workshops
    workers_24 There are 2743 people working in three workshops. In the second workshop works 140 people more than in the first and in third works 4.2 times more than the second one. How many people work in each workshop?
  8. Isosceles triangle
    triangle2_3 The leg of the isosceles triangle is 5 dm, its height is 20 cm longer than the base. Calculate base length z.
  9. Equation
    calculator_2 Equation ? has one root x1 = 8. Determine the coefficient b and the second root x2.
  10. Theorem prove
    thales_1 We want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started?
  11. Algebra
    parabol_3 X+y=5, find xy (find the product of x and y if x+y = 5)
  12. Quadratic equation
    kvadrat_2 Find the roots of the quadratic equation: 3x2-4x + (-4) = 0.
  13. Discriminant
    Quadratic_equation_discriminant Determine the discriminant of the equation: ?
  14. Elimination method
    rovnice_1 Solve system of linear equations by elimination method: 5/2x + 3/5y= 4/15 1/2x + 2/5y= 2/15
  15. Holidays - on pool
    pool_4 Children's tickets to the swimming pool stands x € for an adult is € 2 more expensive. There was m children in the swimming pool and adults three times less. How many euros make treasurer for pool entry?
  16. Roots
    parabola Determine the quadratic equation absolute coefficient q, that the equation has a real double root and the root x calculate: ?
  17. Legs
    rak Cancer has 5 pairs of legs. The insect has 6 legs. 60 animals have a total of 500 legs. How much more are cancers than insects?