# A skating board rocks back and forth on a wooden cylinder. The cylinder has a radius of 6 inches and a surface area of 590 $i{{n}^{2}}$. Find the height of the cylinder. Consider $\pi $ = 3.14.

(a) 10 in

(b) 9.65 in

(c) 9 in

(d) 9.5 in

Last updated date: 28th Mar 2023

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Answer

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Hint: To solve this question, one must know the formula of the surface area of a cylinder which we study in surface area and volume. Assume a variable r which will represent the radius of a cylinder and assume a variable h which will represent the height of the cylinder. The surface area of this cylinder is given by the formula $S=2\pi r\left( r+h \right)$. Using this formula, we can solve this question.

Complete step-by-step answer:

Before proceeding with the question, we must know the formula that will be required to solve this question.

In surface area and volumes, for a cylinder of radius r and height h, the surface area of the cylinder is given by the formula $S=2\pi r\left( r+h \right)$ . . . . . . . . . . . . (1)

In this question, we are given a cylinder of radius 6 inches and having a surface area of 590 $i{{n}^{2}}$. We are required to find the height of this cylinder.

Let us consider h to be the height of the cylinder. Using formula (1), we get,

$\begin{align}

& S=2\pi r\left( r+h \right) \\

& \Rightarrow 590=2\left( 3.14 \right)\left( 6 \right)\left( 6+h \right) \\

& \Rightarrow 590=37.68\left( 6+h \right) \\

& \Rightarrow h+6=\dfrac{590}{37.68} \\

& \Rightarrow h+6=15.65 \\

& \Rightarrow h=9.65 in \\

\end{align}$

Hence, the answer is option (c).

Note: There is a possibility that one may use the formula for the curved surface area of the cylinder i.e. $S=2\pi rh$. But since in the question, it is just written that we have to find the surface area, we will assume it as the total surface area which is given by the formula $S=2\pi r\left( r+h \right)$.

Complete step-by-step answer:

Before proceeding with the question, we must know the formula that will be required to solve this question.

In surface area and volumes, for a cylinder of radius r and height h, the surface area of the cylinder is given by the formula $S=2\pi r\left( r+h \right)$ . . . . . . . . . . . . (1)

In this question, we are given a cylinder of radius 6 inches and having a surface area of 590 $i{{n}^{2}}$. We are required to find the height of this cylinder.

Let us consider h to be the height of the cylinder. Using formula (1), we get,

$\begin{align}

& S=2\pi r\left( r+h \right) \\

& \Rightarrow 590=2\left( 3.14 \right)\left( 6 \right)\left( 6+h \right) \\

& \Rightarrow 590=37.68\left( 6+h \right) \\

& \Rightarrow h+6=\dfrac{590}{37.68} \\

& \Rightarrow h+6=15.65 \\

& \Rightarrow h=9.65 in \\

\end{align}$

Hence, the answer is option (c).

Note: There is a possibility that one may use the formula for the curved surface area of the cylinder i.e. $S=2\pi rh$. But since in the question, it is just written that we have to find the surface area, we will assume it as the total surface area which is given by the formula $S=2\pi r\left( r+h \right)$.

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