A cube of side 11 cm is melted and converted into a solid cylinder. It is found that the height of the cylinder so formed is 7 times the length of the rectangle whose width is 1.5 cm and perimeter 4 cm. Find the radius of the cylinder?

Let’s break this down step by step:

1. Volume of the Cube : The volume of a cube is given by \( \text{Volume} = a^3 \), where \( a \) is the side length of the cube. Here, \( a = 11 \) cm. So, the volume of the cube is \( 11^3 \) cubic cm.

2. Volume of the Cylinder : The volume of a cylinder is given by \( \text{Volume} = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. We are given that the height of the cylinder is 7 times the length of the rectangle. The length of the rectangle is not explicitly given, but we can find it using the given information about the width and perimeter of the rectangle.

3. Finding the Length of the Rectangle : The perimeter of a rectangle is given by \( 2(l + w) \), where \( l \) is the length and \( w \) is the width. We are given that the width \( w = 1.5 \) cm and the perimeter is 4 cm. Therefore, \( 2(l + 1.5) = 4 \). Solving for \( l \), we get \( l = \frac{4}{2} – 1.5 = 0.5 \) cm.

4. Height of the Cylinder : Since the height of the cylinder is 7 times the length of the rectangle, the height is \( 7 \times 0.5 = 3.5 \) cm.

5. Volume of the Cylinder (Continued) : Now that we have the height of the cylinder as 3.5 cm, we can find its volume using the formula \( \pi r^2 h \).

6. Equating Volumes : Since the cube is melted to form the cylinder, the volume of the cube should be equal to the volume of the cylinder. Setting these two volumes equal to each other, we can solve for the radius \( r \) of the cylinder.

Let’s calculate the radius \( r \) of the cylinder using the given information.

Given:
Side length of cube, \(a = 11\) cm
Width of rectangle, \(w = 1.5\) cm
Perimeter of rectangle, \(P = 4\) cm

1. Volume of Cube : \(V_{\text{cube}} = a^3 = 11^3\) cm³.

2. Length of Rectangle : Perimeter of rectangle, \(P = 2(l + w)\). We have \(P = 4\) and \(w = 1.5\). Solve for \(l\):

\[4 = 2(l + 1.5) \]
\[2 = l + 1.5 \]
\[l = 0.5\] cm.

3. Height of Cylinder : Height of cylinder, \(h = 7l = 7 \times 0.5\) cm.

4. Volume of Cylinder : Volume of cylinder, \(V_{\text{cylinder}} = \pi r^2 h\).

Since the cube is melted and converted into the cylinder, their volumes are equal:

\[11^3 = \pi r^2 \times 7 \times 0.5\]

\[1331 = 3.5 \pi r^2\]

\[r^2 = \frac{1331}{3.5\pi}\]

\[r = \sqrt{\frac{1331}{3.5\pi}}\]

To find the radius of the cylinder, we first calculate the value inside the square root:

\[ r = \sqrt{\frac{1331}{3.5\pi}} \]

\[ r = \sqrt{\frac{1331}{3.5 \times 3.14159}} \]

\[ r = \sqrt{\frac{1331}{10.99265}} \]

\[ r = \sqrt{121} \]

\[ r = 11 \text{ cm} \]

Therefore, the radius of the cylinder is 11 cm.