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Math  /  Geometry

Question8. The underside of a concrete bridge forms a parabolic arch that is 40 m wide and 16 m tall at its centre. What is the height of the underside of the bridge exactly 5 m from the axis of symmetry? ∴\therefore the helght of the undesside of the bridge 5 m foom Aos is 11 (to the lef) or 21 (to the right).

Studdy Solution

STEP 1

1. The parabolic arch is symmetric about its vertical axis.
2. The vertex of the parabola is at the highest point of the arch, which is 16 m tall.
3. The width of the arch at the base is 40 m.

STEP 2

1. Determine the equation of the parabola.
2. Substitute the horizontal distance from the axis of symmetry into the equation.
3. Calculate the height at the specified distance.

STEP 3

Determine the equation of the parabola:
Since the parabola is symmetric and has its vertex at the origin of the coordinate system, we can place the vertex at (0,16)(0, 16). The parabola opens downward, and its equation can be written in vertex form as:
y=a(x−h)2+k y = a(x - h)^2 + k
Given h=0 h = 0 and k=16 k = 16 , the equation becomes:
y=a(x)2+16 y = a(x)^2 + 16
To find a a , use the fact that the parabola is 40 m wide at the base, meaning when x=±20 x = \pm 20 , y=0 y = 0 .
Substitute (20,0) (20, 0) into the equation:
0=a(20)2+16 0 = a(20)^2 + 16
Solve for a a :
0=400a+16 0 = 400a + 16 400a=−16 400a = -16 a=−16400 a = -\frac{16}{400} a=−125 a = -\frac{1}{25}
Thus, the equation of the parabola is:
y=−125x2+16 y = -\frac{1}{25}x^2 + 16

STEP 4

Substitute the horizontal distance from the axis of symmetry into the equation:
We need to find the height at x=5 x = 5 .
Substitute x=5 x = 5 into the equation:
y=−125(5)2+16 y = -\frac{1}{25}(5)^2 + 16

STEP 5

Calculate the height at the specified distance:
y=−125(25)+16 y = -\frac{1}{25}(25) + 16 y=−1+16 y = -1 + 16 y=15 y = 15
The height of the underside of the bridge exactly 5 m from the axis of symmetry is:
15 m \boxed{15 \text{ m}}

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