Question

An arch is in the shape of a parabola whose axis is vertically down words and measures 40 mts across its bottom on the ground. Its heighest point is 12 mts. The measure of the
horizontal beam across its cross
section at a height of 9 mts is​

Answer 1

The length of horizontal beam = 10 miters

Step-by-step explanation:

We assume that the axis of the parabola is the y−axis.

Hence, we have $x^2=-4a(y-k)$ .....(1)

We take the negative sign due to vertically downwards axis.

Now the highest point is at 12 mitres.

So, k=12

Put in (1), we get.

$x^2=-4a(y-12)$

Now, parabola measures 40 mitres across its bottom on the ground.

Hence,

By symmetry (20,0) will lie on the parabola.

So, we put x = 20 in the above equation.

$20^2=-4a(0-12)$

48a=400

$a=\dfrac{25}{3}$

Thus; $x^2= \dfrac{-25}{3}(y-12)$

Now put y =9

$x^2=\dfrac{-25}{3}\times(9−12)$

$x^2=\dfrac{-25}{3}\times(-3)$

$x^2= \sqrt{25}$

$x=\pm5$

Hence, the length of horizontal beam=5+5=10 miters