Question

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Answer 1

» Question :

The curved-surface of a cylinder is 2π(y² - 7y + 12) and its radius is (y - 3). Find the height of the cylinder.

» To Find :

The height of the cylinder.

» Given :

• Curved-surface area = $2\pi (y^{2} - 7y + 12$

• Radius = $(y - 3)$

» We Know :

Curved surface of a cylinder :

$\mathtt{\underline{\boxed{CSA = 2 \pi rh}}}$

» Concept :

In the given curved-surface area ,the equation i.e, (y² - 7x + 12) ,can be solved into factors for ease of calculation .

Expression :

$\mathtt{y^{2} - 7y + 12}$

By using ,the middle-splitting factor theorem ,we get :

$\mathtt{\Rightarrow y^{2} - (3 + 4)y + 12}$

$\mathtt{\Rightarrow y^{2} - 3y - 4y + 12}$

$\mathtt{\Rightarrow y(y - 3) - 4(y - 3)}$

Taking the common factor (y - 3) ,we get :

$\mathtt{\Rightarrow (y - 3)(y - 4)}$

$\mathtt{\therefore (y^{2} - 7y + 12) = (y - 3)(y - 4)}$

Hence ,the curved-surface area can be written as :

$\sf{\boxed{2\pi[(y - 3)(y - 4)]}}$

» Solution :

• Curved-surface area = 2π[(y - 3)(y - 4)]

• Radius = (y - 3)

Formula :

$\mathtt{\underline{\boxed{CSA = 2 \pi rh}}}$

Putting the value of the radius and the curved surface area in the formula ,we get :

$\mathtt{\Rightarrow 2\pi[(y - 3)(y - 4)] = 2 \pi \times (y - 3)h}$

$\mathtt{\Rightarrow \dfrac{2\pi[(y - 3)(y - 4)]}{2 \pi} = (y - 3)h}$

$\mathtt{\Rightarrow \dfrac{\cancel{2\pi}[(y - 3)(y - 4)]}{\cancel{2\pi}} = (y - 3)h}$

$\mathtt{\Rightarrow (y - 3)(y - 4) = (y - 3)h}$

$\mathtt{\Rightarrow \dfrac{(y - 3)(y - 4)}{(y - 3)} = h}$

$\mathtt{\Rightarrow \dfrac{\cancel{(y - 3)}(y - 4)}{\cancel{(y - 3)}} = h}$

$\mathtt{\Rightarrow (y - 4) = h}$

Hence ,the height of the cylinder is (y - 4).

» Additional information :

• Total-surface area of a cylinder = 2πr(h + r)

• Volume of cylinder = πr²h

• Surface area of the Cuboid = 2(lb + lh + bh)

• Curved surface area of rectangle = 2(l + b)h

Where ,

h = height

r = radius

l = length

b = breadth

Answer 2

Step-by-step explanation:

$\bf{\underline{\underline\red{Given:-}}}$

• The curved surface area of cylinder = 2π(y²- 7y + 12).

• Radius of cylinder = (y - 3)

$\bf{\underline{\underline\blue{To\:Find:-}}}$

• The height of the cylinder.

$\bf{\underline{\underline\green{Solution:-}}}$

As we know that:-

Curved surface area (CSA) of cylinder is given by the formula:-

$\boxed{ \rm \leadsto CSA\: of \: cylinder \: = 2\pi rh}$

Here:-

• r = radius = (y - 3)

• h = height

• CSA = 2π(y² - 7y + 12)

Substituting the values:-

$\rm \implies2\pi( {y}^{2} - 7y + 12) \: = 2\pi rh$

$\rm \implies2 \not\pi( {y}^{2} - 7y + 12) \: = 2 \not\pi (y - 3)h$

$\rm \implies\not2( {y}^{2} - 7y + 12) \: = \not2 (y - 3)h$

$\rm \implies {y}^{2} - 7y + 12 \: = (y - 3)h$

$\rm \implies {y}^{2} - 3y - 4y + 12 \: = (y - 3)h$

$\rm \implies y (y- 3) - 4(y - 3)\: = (y - 3)h$

$\rm \implies (y- 3) (y - 4)\: = (y - 3)h$

$\rm \implies \dfrac{(y- 3) (y - 4)}{(y - 3)}\: = h$

$\rm \implies (y - 4) = h$

$\rm \implies h= (y - 4)$

$\boxed{ \rm \therefore The \: height \: of \: the \: cylinder \: =( y - 4)}$