Mensuration

Step 1: {Relate volume and surface area of the cube}
The volume of the cube is: where is the length of an edge. Differentiating with respect to , we get: Substitute : Solve for : Step 2: {Find the rate of change of surface area}
The surface area of the cube is: Differentiating with respect to , we get: Substitute and : Conclusion: The surface area of the cube is increasing at .

Let the dimensions of the rectangle be and , where is the shorter side. The perimeter condition is: Thus, . When the rectangle is swept along a circular rotation with the shorter side fixed as the axis, the volume of the solid formed is given by the formula for the volume of a cylinder: Now, to find the maximum volume, differentiate with respect to : Set to find the critical points: Thus, or , so . Substitute into the perimeter equation to find : Thus, the dimensions of the rectangle are cm by cm.

The volume of a cylinder is given by the formula: where is the radius and is the height (altitude) of the cylinder. To find the rate of change of volume with respect to time, we differentiate both sides of the equation with respect to time : Using the product rule: We are given that , , , and . Substituting these values into the equation: Thus, the rate of change of the volume is:

Let the length and width of the rectangle be denoted by and respectively. The perimeter is fixed, so: The area of the rectangle is given by: Now, express in terms of : Substitute this into the area equation: This is a quadratic function, and we need to find the value of that maximizes . To do this, we take the derivative of with respect to : Set to find the critical point: Since , we also have: Thus, the length and width are equal, and the rectangle is a square. This shows that for a fixed perimeter, the square has the greatest area.

Let the length and breadth of the box be m and the height be m. The surface area consists of the area of the base and the area of the four sides: Now, the volume of the cuboid is given by: Since is constant, we can express in terms of and : Substitute this into the surface area expression: