The area of a square can be given in terms of its sides as

However, we're told that the horizontal and vertical sides have different rates of change. It would be helpful to rewrite the equation in light of this fact:

.

Now, the rates of change of the parameters can be related by taking the derivative of each side of the equation with respect to time. Be sure to take the derivative with respect to each variable and sum the results.

Now, we can utilize our rates of change . Recall that :

The hypotenuse of a right triangle can be related to the triangle's legs via the Pythagorean Theorem:

We're told that :

Now let's return to the Pythagorean Theorem:

We can use it to relate rates of change by taking the derivative of each side of the equation with respect to time:

We're told that , so we can in turn solve for :

The area of a rectangle in terms of its sides is

For the sake of this problem, let  denote the shorter side and  the longer.

This formula can also be used to relate rates of change. We need only take the derivative of each side of the equation with respect to time. Be sure to take the derivative with respect to each variable:

We're given some rates of change in the problem statement. Primarily

 and 

Combining this with the lengths  and , we can find the rate of change of the smaller side:

The surface area of a cube is given in terms of its sides as

This relation can be used to also relate rates of change. This can be done by taking the derivative of each side of the equation with respect to time:

We are given the rate of change of the sides  and the lengths of the sides . Thus the rate of growth of the surface area is

The area of a trapezoid is given by the formula

Let a represent the short side and b the long side.

For this problem, while not necessary, it may make things easier to expand this equation

Now, we can relate rates of change by taking the derivative of each side of the equation with respect to time:

Now recall that the height is constant. This means that , which shortens the equation

The surface area of a rectangular prism is given by the formula:

The rate of change of parameters can be found by taking the derivative of both sides of the equation with respect to time:

Now, we've been given some rates of change, namely . Since the height is constant , and the equation reduces to

Plugging in our other known values , we can solve for the rate of change of the surface area

Although we are told that the sides of the cube have different rates of growth (meaning it won't stay a cube for long), at the time of this problem, since the shape is a cube it means all of the sides are equal:

The volume of a cube is given by the expression:

We'll be using this to find the current lengths of the sides

Now, rewrite the volume equation in terms of the three sides:

The rate of change of the volume can be found by taking the derivative of each side of the equation with respect to time. Remember to take the derivative of the right side with respect to each variable:

With the three rates of change , , and  and the known length of the sides, the rate of change is

For this problem we're given the initial radius of a sphere and the rate of change of this radius. It will be useful to find an equation for the radius itself, however.

The rate, we are told, is .

To find an expression for the radius, we need to integrate this function.

To find C, the constant of integration, utilize the initial condition:

So our radius expression is

Now to find the maximum radius, take the limit as time goes towards infinity:

The volume of a sphere is given by the equation:

Therefore, the maximum volume is

The area of a rectangle in relation to its sides is given by the formula

This rule holds even when the the sides are expressed in more complicated terms such as those of this problem.

Since  and 

The rate of change of the area can be found by taking the derivative of both sides of the equation with respect to time.

Recall that the rule of derivation for an exponential:

, where  is any function in the exponent.

For this problem, the derivative is then

This is the formula for the rate of growth of the rectangle's area.

The area of a square is given in terms of its sides as

For an area of 

The area equation can also be used to find rates of change; take the derivative of each side of the equation with respect to time:

We're told that 

Therefore