Question

Trigonometric Functions Question (Area underneath a curve). Help please?

An arched window has a base length of 4m and a height of 2m. The arch is to be either an arc of a parabola or a half-period of a sine curve.

(a) If the arch is the arch of a parabola, the equation of the curve is `f(x)=ax(4-x)`. Show that the value of `a` is `1/2`.

(b) If the arch is sinusoidal, the equation is the form `g(x)=Asin((pix)/4)`. Find the value of `A`.

(c) Calculate the area for each window design and hence, determine which one has less area.

Thanks!

Answer 1

Area of f(x) : `5.33 m^2`
Area of g(x) : `5.09 m^2`

The sinusoidal design has lesser area.

Explanation:

The height of the parabolic design is at the vertex, with the coordinate `(2,2)`:
`2 = a*2*(4-2)`
`a = frac{1}{2}`
The height of the sinusoidal design is the amplitude, so `A = 2`
Now the area under the graph: (I assume you know about integral)
The parabolic design:
Area = `int_0^4 1/2*x*(4-x)d\x = 5.33m^2`
The sinsoidal design:
Area = `int_0^4 2*sin(pi*x/4)d\x = 5.09m^2`

Therefore, the parabolic design is bigger.