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

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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!

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

(b) If the arch is sinusoidal, the equation is the form

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

Thanks!

##### 1 Answer

Area of f(x) :

Area of g(x) :

The sinusoidal design has lesser area.

#### Explanation:

The height of the parabolic design is at the vertex, with the coordinate

The height of the sinusoidal design is the amplitude, so

Now the area under the graph: (I assume you know about integral)

The parabolic design:

Area =

The sinsoidal design:

Area =

Therefore, the parabolic design is bigger.