Optimization

The distance between the point on the curve and the soldier at is given by the distance formula: Substitute into the distance formula: We want to minimize this distance, so we minimize to avoid dealing with the square root. Define the function: To minimize , take the derivative and set it to zero: Factor the equation: Solving this equation, we find that . Now substitute into the distance formula: Thus, the point on the curve is . The nearest distance is: Final Answer: The nearest distance is .

Step 1: Define variables. Let the two numbers be and , so The function to minimize is: Step 2: Express in terms of one variable. Using , we get: Step 3: Differentiate and find the minimum. Setting : Thus, the minimum sum of squares occurs when .

For a cone, volume . Using the slant height : Substitute and differentiate to maximize . Solve for where . The result is:

Plot the constraints on a graph to form the feasible region. The vertices of the feasible region are determined by solving the intersection points: Substitute the vertices into : The minimum value of is .