Q) The difference between the outer and inner radii of a hollow right circular cylinder of length 14 cm is 1 cm. If the volume of the used in making the cylinder is 176 cm³, find the outer and inner radii of the cylinder.

Ans.

Let’s draw the diagram for the given question: The difference between the outer and inner radii of a hollow right circular cylinder of length 14 cm is 1 cm. CBSE 2024

Here, Cylinder has outer radii as R_o and inner radii as R_i, height 14 cm.

We know that the volume of the Cylinder, V_c = \pi (r)^2 h

Given that h = 14 cm

Here, outer volume V_o = \pi (R_o)^2 (14)

And  Inner volume V_i = \pi (R_i)^2 (14)

From the diagram, it is clear that the difference of the outer and inner volume is the volume of the metal. It is given that metal volume is 176 cm 3

Therefore, \pi (R_o)^2 (14) - \pi (R_i)^2 (14) = 176

\Rightarrow \pi (14) [(R_o)^2 - (R_i)^2] = 176

\Rightarrow \frac{22}{7}(14) [(R_o + R_i)(R_o - R_i)] = 176

Given that R_o - R_i = 1 ……… (i)

\Rightarrow 44[(R_o + R_i)(1)] = 176

\Rightarrow (R_o + R_i) = 4 ….. (ii)

By adding Equation (i) and (ii), we get:

(R_o - R_i) + (R_o + R_i) = 1 + 4

2 R_o = 5

R_o = \frac{5}{2}

Substituting value of R_o in equation (ii), we get:

R_o + R_i = 4

\Rightarrow \frac{5}{2} + R_i = 4

\Rightarrow R_i = 4 - \frac{5}{2}

\Rightarrow R_i = \frac{3}{2}

Therefore, the outer radii of cylinder is \frac{5}{2} cm and inner radii is \frac{3}{2} cm.

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