#### To determine

**To find:**

The value of c

#### Answer

3227

#### Explanation

**1) Given:**

Equation of the parabola is y=8x-27x3

**2) Calculation:**

Let x1,c and x2,c be the intersection points of y=c and y=8x-27x3.

Let the area of the region below line y=c be R1 and the area of the region above line y=c be R2.

We need to find value of c where area of region R1 and area of region R2 is the same.

So assume that both areas are the same.

Setting both area in terms of integration, it becomes

∫0x1c-8x-27x3 dx= ∫x1x28x-27x3-cdx

That is,

∫0x1c-8x+27x3 dx=∫x1x28x-27x3-cdx

Integrating,

cx-8x22+27x44x10= 8x22-27x44-cxx2x1

Substituting the limits,

cx1-8x122+27x144=8x222-27x244-cx2- 8x122-27x144-cx1

cx1-8x122+27x144=8x222-27x244-cx2-8x122+27x144+ c x1

Cancelling out common terms from both sides,

8x222-27x244-cx2=0

Simplifying,

4x22-27x244-cx2=0

Now, we need to find intersection points of y=c and y=8x-27x3. At intersection

c=8x-27x3

Since the point x2,c is one of the intersection points, substitute it in c=8x-27x3

c=8x2-27x23

Substituting this in the above simplified equation,

4x22-27x244-8x2-27x23x2=0

4x22-27x244-8x22+27x24=0

81x244- 4x22=0

Factoring out x22,

x22814x22-4=0

Factoring it out,

x2292x2-292x2+2=0

Therefore,

x2=0 or x2=±49 From the figure, it can be seen that x2 cannot be 0 or negative.

Therefore, x2=49

As,

c=8x2-27x23

c=849-27493

After simplifying,

c=329-6427=3227

**Conclusion:**

The value of c is 3227