An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?
Solution and Explanation
The origin of the coordinate plane is taken at the vertex of the arch in such a way that its vertical axis is along the positively-axis.
This can be diagrammatically represented as
The equation of the parabola is of the form \(x^ 2 = 4ay\) (as it is opening upwards).
It can be clearly seen that the parabola passes through point \((\frac{5}{2}, 10)\)
\((\frac{5}{2})^2 = 4a(10)\)
\(4a = \frac{25}{(4\times4\times10)}\)
\(⇒ a =\frac{ 5}{32}\)
Therefore, the arch is in the form of a parabola whose equation is \(x^2 = \frac{5}{8} (2)\)
When \(y = 2 m, x^2 = \frac{5}{8} \times(2)\)
\(⇒ x^2 = \frac{5}{4}\)
\(x = \sqrt{(\frac{5}{4})} = \sqrt{\frac{5}{2}}\)
\(∴ AB = 2 \times \sqrt{\frac{5}{2}}m = \sqrt{5}m = 2.23m\)(approx.)
Hence, when the arch is 2 m from the vertex of the parabola, its width is approximately 2.23 m.
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Concepts Used:
Parabola
Parabola is defined as the locus of points equidistant from a fixed point (called focus) and a fixed-line (called directrix).
Standard Equation of a Parabola
For horizontal parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A,
- Two equidistant points S(a,0) as focus, and Z(- a,0) as a directrix point,
- P(x,y) as the moving point.
- Let us now draw SZ perpendicular from S to the directrix. Then, SZ will be the axis of the parabola.
- The centre point of SZ i.e. A will now lie on the locus of P, i.e. AS = AZ.
- The x-axis will be along the line AS, and the y-axis will be along the perpendicular to AS at A, as in the figure.
- By definition PM = PS
=> MP2 = PS2
- So, (a + x)2 = (x - a)2 + y2.
- Hence, we can get the equation of horizontal parabola as y2 = 4ax.
For vertical parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A
- Two equidistant points, S(0,b) as focus and Z(0, -b) as a directrix point
- P(x,y) as any moving point
- Let us now draw a perpendicular SZ from S to the directrix.
- Then SZ will be the axis of the parabola. Now, the midpoint of SZ i.e. A, will lie on P’s locus i.e. AS=AZ.
- The y-axis will be along the line AS, and the x-axis will be perpendicular to AS at A, as shown in the figure.
- By definition PM = PS
=> MP2 = PS2
So, (b + y)2 = (y - b)2 + x2
- As a result, the vertical parabola equation is x2= 4by.