Question

State the following statement is true or false.
A parabolic arch has a height of 18 meters and spans 24 meters. Then the height of the arch at 8 meters from the centre of the span is equal to 10.
a. true
b. false

Answer 1

Hint: First we need to draw the figure according to the given conditions and heights. We can consider two points on the parabola. Then from one point, we can find the value of a, then using the value of a and with the coordinates of another point, we can evaluate the value of the height.

Complete step by step solution:

We have a parabolic arc as shown in the figure.
Now, it is said that the parabolic arch has a height of 18 meters and it has a span of 24 meters in total. We have our axis as CE and our centre in D, now we have our span is 24 meters, so, if we take E as a point on the axis, we get, \[EF = GE = 12\] meters. And also, the arch has a height of 18 meters. Now we consider the point D where the height is 8 meters and AB is intersecting the axis.
Let, DE=h meters. So, CD= \[18 - h\] meters.
 By taking the equation of parabola in standard form \[{y^2} = 4ax,\]we get coordinates of F as \[\left( {18,12} \right)\] and it should satisfy the equation of the parabola.
Hence, \[{12^2} = 4a \times 18\]
On simplification we get,
\[ \Rightarrow 144 = 4a \times 18\]
On Dividing equation by 18, we get,
\[ \Rightarrow 4a = 8\].....(1)
Now, the height of the arc (distance from centre) for a point which is at a distance of 8 meters from the span (y-coordinate in the figure) be h.
Then, the coordinates of one point on the parabola become A \[\left( {18 - h,8} \right)\], and it should also the equation of the parabola, we have,
Hence, \[{8^2} = 4a \times \left( {18 - h} \right)\]
On simplification we get,
\[ \Rightarrow 64 = 4a \times \left( {18 - h} \right)\]
From equation 1, we get,
\[ \Rightarrow 64 = 8 \times \left( {18 - h} \right)\]
On dividing the equation by 8 we get,
\[ \Rightarrow 18 - h = 8\]
Hence, on simplification we get,
\[ \Rightarrow h = 10\]
Hence, we can say that A parabolic arch has a height of 18 meters and span 24 meters. Then the height of the arch at 8 meters from the centre of the span is equal to 10.
So, the given statement is true.

Hence, option (a) is correct.

Note: You should always first draw the figure and mark the points, for better understanding.
We have the properties of a parabola as,
1) The eccentricity of any parabola is 1.
2) The parabola is symmetric about its axis.
3) The axis is perpendicular to the directrix.
4) The axis passes through the vertex and the focus.
5) The tangent at the vertex is parallel to the directrix.
6) The vertex is the midpoint of the focus and the point of intersection of directrix and axis.
7) Tangents drawn to any point on the directrix are perpendicular.