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!

1 Answer
Oct 3, 2017

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.