An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end

Solution:

Since the height and width of the arc from the centre is 2 m and 8 m respectively,

it is clear that the length of the major axis is 8 m, while the length of the semi-minor axis is 2 m.

The origin of the coordinate plane is taken as the centre of the ellipse, while the major axis is taken along the x-axis.

Hence, the semi-ellipse can be diagrammatically represented as

The equation of the semi-ellipse will be of the form x²/a² + y²/b² = 1 y ≥ 0, where a is the semi-major axis.

Accordingly,

⇒ 2a = 8 ⇒ a = 4 and ⇒ b = 2

Therefore, the equation of the semi-ellipse is

x²/16 + y²/ = 1, y ≥ 0 ....(1)

Let A be a point on the major axis such that AB = 1.5 m.

Now, Draw AC ⊥ OB .

OA = (4 - 1.5) m = 2.5 m

The x-coordinate of point C is 2.5

On substituting the value of x = 2.5 in equation (1), we obtain

⇒ (2.5)²/16 + y²/4 = 1

⇒ 6.25/16 + y²/4 = 1

⇒ y² = 4 (1 - 6.25/16)

⇒ y² = 4 (9.75/16)

⇒ y² = 2.4375

⇒ y = 1.56 (approx.)

Therefore, AC = 1.56 m

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NCERT Solutions Class 11 Maths Chapter 11 Exercise ME Question 4

An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end

Summary:

The height of the arch at a point 1.5 m from one end is approximately 1.56 m