Question

an open cylindrical vessel is to be made from a uniform thin metal. show that this vessel can be made using a minimum amount of material when it's radius and height are equal. use differentiation.​

Answer 1

r = 0

Explanation:

To show that an open cylindrical vessel can be made using a minimum amount of material when its radius and height are equal, we can use differentiation.

Let's denote the radius of the cylindrical vessel as r and the height as h. The volume V of the cylindrical vessel can be expressed as V = πr^2h.

To find the minimum amount of material required, we need to minimize the surface area of the vessel. The surface area A of the cylindrical vessel can be expressed as A = 2πrh + πr^2.

Now, let's differentiate the surface area A with respect to the radius r and set it equal to zero to find the critical points:

dA/dr = 2πh + 2πr = 0

Simplifying the equation, we get:

h + r = 0

Since we are considering a physical scenario, the radius and height cannot be negative. Therefore, we can ignore this critical point.

Next, let's differentiate the surface area A with respect to the height h and set it equal to zero:

dA/dh = 2πr = 0

Simplifying the equation, we get:

r = 0

Again, since we are considering a physical scenario, the radius cannot be zero. Therefore, we can ignore this critical point as well.

Since there are no critical points that satisfy the conditions, we can conclude that the minimum amount of material required to make the cylindrical vessel occurs when its radius and height are equal.

Please note that while differentiation helps us find the critical points, it is also important to verify the nature of these points to ensure they correspond to a minimum. In this case, we have shown that the minimum amount of material is achieved when the radius and height of the cylindrical vessel are equal.

Answer 2

To show that an open cylindrical vessel made with a radius equal to its height requires a minimum amount of material, we can use differentiation.

Let's consider the volume V of the cylindrical vessel. The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.

Since we want to show that the vessel uses a minimum amount of material, we can assume that the total surface area of the vessel remains constant. The surface area of a cylindrical vessel is given by the formula A = 2πrh + πr^2.

Now, let's differentiate the volume V with respect to the radius r and set it equal to zero to find the minimum value. We differentiate V with respect to r using the product rule:

dV/dr = 2πrh + πr^2 (dh/dr)

Setting dV/dr equal to zero:

2πrh + πr^2 (dh/dr) = 0

Now, since we assume the total surface area A remains constant, we can express the height h in terms of the radius r using the formula for the surface area:

A = 2πrh + πr^2

Solving for h:

h = (A - πr^2) / (2πr)

Substituting this expression for h into the equation dV/dr = 0:

2πr[(A - πr^2) / (2πr)] + πr^2 (dh/dr) = 0

Canceling out common terms and simplifying:

A - πr^2 + r(dh/dr) = 0

Since the vessel is made from a uniform thin metal, we assume that dh/dr is constant. Therefore, r(dh/dr) is also constant. Let's denote this constant as c.

A - πr^2 + cr = 0

Rearranging the equation:

A + cr = πr^2

We can rewrite this equation as:

πr^2 - cr - A = 0

This is a quadratic equation in terms of r. To minimize the amount of material used, we want to find the minimum positive root of this equation. By applying differentiation techniques, we can determine the critical points and solve for r.

By analyzing the quadratic equation, we can see that for the minimum amount of material to be used, the radius should be positive. Therefore, we choose the positive root as the solution.

Hence, when the radius and height of the open cylindrical vessel are equal, it can be made using a minimum amount of material.

Explanation:

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