To optimize the cone's surface area with a fixed volume of 1500 cm³, we need to minimize the surface area while ensuring the volume is constant. Let's break down the process.

1. Volume and Surface Area Formulas

For a cone, the following formulas are used:

• Volume of the cone $V$: $V = \frac{1}{3} \pi r^2 h$ Where:

  • $r$ = radius of the base,
  • $h$ = height of the cone.

• Surface Area of the cone $A$: $A = \pi r^2 + \pi r l$ Where:

  • $r$ = radius of the base,
  • $l$ = slant height, which is given by: $l = \sqrt{r^2 + h^2}$

2. Given Conditions

• Fixed volume of the cone: $V = 1500 \, \text{cm}^3$
  • $\frac{1}{3} \pi r^2 h = 1500$ From this, we can express $h$ in terms of $r$: $h = \frac{4500}{\pi r^2}$

3. Surface Area as a Function of $r$

Now, substitute $h$ in the surface area formula: $A = \pi r^2 + \pi r \sqrt{r^2 + h^2}$ Substitute the expression for $h$: $A = \pi r^2 + \pi r \sqrt{r^2 + \left(\frac{4500}{\pi r^2}\right)^2}$ This is the surface area expressed as a function of $r$.

4. Optimization

To find the value of $r$ that minimizes the surface area, we take the derivative of $A(r)$ with respect to $r$, set it equal to 0, and solve for $r$.

Let's proceed with the optimization:

Step 1: Differentiate $A(r)$ with respect to $r$.

Step 2: Solve for $r$.

Let me perform the differentiation and solve for $r$. sympy import symbols, pi, sqrt, diff, Eq, solve

Define the variables

r = symbols('r')

Given volume

V = 1500

h in terms of r

h = 4500 / (pi * r**2)

Surface area equation

A = pi * r2 + pi * r * sqrt(r2 + h**2)

Differentiate the surface area with respect to r

A_prime = diff(A, r)

Solve A'(r) = 0 to find the critical points

critical_points = solve(Eq(A_prime, 0), r) critical_pointsThe critical points from the derivative of the surface area with respect to $r$ are:

1. $r = \frac{5 \cdot 2^{1/6} \cdot 3^{2/3}}{\pi^{1/3}}$
2. Two complex values that are not physically meaningful for this optimization problem.

We can disregard the complex solutions and focus on the real solution for $r$.

Let's now compute this real value for $r$. Calculate the real value of r real_r = 5 * 2**(1/6) * 3**(2/3) / pi**(1/3) real_r.evalf()The radius $r$ that minimizes the surface area of the cone, while maintaining a fixed volume of 1500 cm³, is approximately 7.97 cm.

Would you like to go over any more details, or need help with any further steps? Here are a few related questions you might find interesting:

1. How can I calculate the corresponding height $h$ for this optimized radius?
2. What is the surface area at this optimized value of $r$?
3. Can this optimization be applied to cones with different fixed volumes?
4. How would the optimization change if the cone had a different shape (e.g., a frustum)?
5. How does the fixed volume constraint influence the slant height $l$?

Tip: When solving optimization problems, it’s always good to check if the critical points lead to a minimum by using the second derivative test or verifying boundary conditions.