A bridge

The bridge over the river has the shape of an arc. The bridge is 10 feet above the water at the center of the river. At 27 feet from the river's edge, the bridge is 9 feet above the water. How wide is the river?

Correct answer:

a =  80 ft

Step-by-step explanation:

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+54y+629  20⋅ (y2−19)/2=y2+54 y+629 9y2−54y−819=0 9=32 54=2⋅33 819=32⋅7⋅13 GCD(9,54,819)=32=9  y2−6y−91=0  a=1;b=−6;c=−91 D=b2−4ac=62−4⋅1⋅(−91)=400 D>0  y1,2​=2a−b±D​​=26±400​​ y1,2​=26±20​ y1,2​=3±10 y1​=13 y2​=−7  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 ft

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Showing 1 comment:
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





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