Among all right circular cones with a slant height of 15 , what are the dimensions (radius and height) that maximize the volume of the cone? The slant height of a cone is the distance from the oute edge of the base to the vertex Let V be the volume of the cone. What is the objective function in terms of the height of the cone, h ? \( \mathrm{V}=\frac{\pi}{3}\left(225 \mathrm{~h}-\mathrm{h}^{3}\right) \) (Type an expression Type an exact answer, using \( \pi \) as needed.) The interval of interest of the objective function is \( [0,225] \) (Simplify your answer Type your answer in interval notation.)

To determine the objective function and the interval of interest for maximizing the volume \( V \) of a right circular cone with a fixed slant height of 15, let's break down the problem step-by-step. ### 1. Relationship Between Slant Height, Radius, and Height For a right circular cone: - **Slant height (\( l \))**: The distance from the edge of the base to the vertex. - **Radius (\( r \))**: The radius of the base. - **Height (\( h \))**: The perpendicular distance from the base to the vertex. These quantities are related by the Pythagorean theorem: \[ l^2 = r^2 + h^2 \] Given that the slant height \( l = 15 \), we have: \[ 15^2 = r^2 + h^2 \implies 225 = r^2 + h^2 \implies r^2 = 225 - h^2 \] ### 2. Volume of the Cone The volume \( V \) of a right circular cone is given by: \[ V = \frac{1}{3}\pi r^2 h \] Substituting \( r^2 = 225 - h^2 \) into the volume formula: \[ V = \frac{1}{3}\pi (225 - h^2) h = \frac{\pi}{3}(225h - h^3) \] Thus, the **objective function** in terms of the height \( h \) is: \[ V(h) = \frac{\pi}{3}(225h - h^3) \] ### 3. Determining the Interval of Interest The height \( h \) must satisfy the condition derived from the Pythagorean relationship: \[ r^2 = 225 - h^2 \geq 0 \implies h^2 \leq 225 \implies h \leq 15 \] Additionally, the height cannot be negative, so: \[ 0 \leq h \leq 15 \] Therefore, the **interval of interest** for the objective function is: \[ [0, 15] \] ### Summary - **Objective Function**: \[ V(h) = \frac{\pi}{3}(225h - h^3) \] - **Interval of Interest**: \[ [0, 15] \] **Final Answer:** The interval of interest is \([0,\ 15]\).