Help please thanks?!

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2 Answers
Feb 2, 2018

See explanation.

Explanation:

The solid's area is equal to the area of a sphere and a side area of a cylinder, so:

(a)

A=4pix^2+2pixyA=4πx2+2πxy

The volume is:

V=4/3pix^3+pix^2yV=43πx3+πx2y

The condition that volume equals pi/6m^3π6m3 allows us to write yy as the function of xx:

4/3pix^3+pix^2y=pi/643πx3+πx2y=π6

4/3x^3+x^2y=1/643x3+x2y=16

8x^3+6x^2y=18x3+6x2y=1

y=(1-8x^3)/(6x^2)y=18x36x2 (b)

Now we can put the calculated yy in the area formula:

A=4pix^2+2pix((1-8x^3)/(6x^2))A=4πx2+2πx(18x36x2)

A=4pix^2+(2pix-16pix^4)/(6x^2)A=4πx2+2πx16πx46x2

A=4pix^2+pi/(3x)-8/3pix^2A=4πx2+π3x83πx2

A=4/3pix^2+pi/(3x)A=43πx2+π3x (c)

To find the minimum area we have to calculate the derivative A'

A'(x)=8/3pix+pi/3*(-1/x^2)

A'(x)=pi/3*(8x-1/x^2)

A'(x)=pi/3*((8x^3-1)/x^2)

A'(x)=0 iff 8x^3-1=0

8x^3=1 => x^3=1/8 => x=1/2

graph{8x^3-1 [-1, 1, -5, 5]}

From the graph of A'(x) we see that the minimum area is reached if x=1/2

The minimum area is:

A(1/2)=4/3pi*(1/2)^2+pi/(3*1/2)

A(1/2)=pi/3+2/3pi=pi (d)

Feb 2, 2018

A.
"Total surface area"="Surface area of cylinder"+"Surface area of two hemispheres"
color(white)("Total surface area")="Surface area of cylinder"+"Surface area of a sphere"
color(white)("Total surface area")=y(2pix)+4pix^2
color(white)("Total surface area")=2pixy+4pix^2

A=2pixy+4pix^2

B.
pi/6=(4pix^3)/3+ypix^2

1/6=(4x^3)/3+yx^2

yx^2=1/6-(4x^3)/3

y=(1/6-(4x^3)/3)/x^2=1/(6x^2)-(4x)/3

C.
A=2pix(1/(6x^2)-(4x)/3)+4pix^2

color(white)(A)=(2pix)/(6x^2)-(2pix(4x))/3+4pix^2

color(white)(A)=pi/(3x)-(8pix^2)/3+4pix^2

color(white)(A)=pi/(3x)+4pix^2-(8pix^2)/3

color(white)(A)=pi/(3x)+(12pix^2-8pix^2)/3

color(white)(A)=pi/(3x)+(4pix^2)/3

D.
A=pi/(3x)+(4pix^2)/3

(dA)/(dx)=d/dx[pi/(3x)+(4pix^2)/3]

color(white)((dA)/(dx))=d/dx[pi/(3x)]+d/dx[(4pix^2)/3]

color(white)((dA)/(dx))=pi/3d/dx[x^(-1)]+(8pix)/3

color(white)((dA)/(dx))=pi/3(-x^(-2))+(8pix)/3

color(white)((dA)/(dx))=-pi/(3x^2)+(8pix)/3

-pi/(3x^2)+(8pix)/3=0

(8pix)/3=pi/(3x^2)

(8x)/3=1/(3x^2)

3x^2(8x)=3

8x^3=1

x^3=1/8

color(white)(x)=1/root(3)(8)

color(white)(A)x=1/2

A=pi/(3(1/2))+(4pi(1/2)^2)/3

color(white)(A)=pi/(3/2)+pi/3

color(white)(A)=(2pi)/3+pi/3

color(white)(A)=picm^2