This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: The volume V of a right circular cone is given by the formula V = (1)/(3) r^2 h\ cm^3, where r is the radius and h is the height. Given the constraint r + h = 5\ cm. Solve for h: h = 5 - r\ cm. Step 2: Substitute h = 5 - r into the volume formula: V = (1)/(3) r^2 (5 - r)\ cm^3. Expand: V = (1)/(3) (5 r^2 - r^3)\ cm^3. Step 3: To find the maximum volume, compute the derivative (dV)/(dr): (dV)/(dr) = (1)/(3) (d)/(dr) (5 r^2 - r^3). First, find the derivative of the inside: (d)/(dr) (5 r^2 - r^3) = 10 r - 3 r^2. So, (dV)/(dr) = (1)/(3) (10 r - 3 r^2)\ cm^3/cm. Step 4: Set the derivative equal to zero to find critical points: (1)/(3) (10 r - 3 r^2) = 0. Since (1)/(3) ≠ 0, solve 10 r - 3 r^2 = 0. Factor: r (10 - 3 r) = 0. Thus, r = 0\ cm or r = (10)/(3)\ cm. Step 5: Check the endpoints and critical points. Domain: 0 < r < 5\ cm (since h > 0). At r 0^+, V 0\ cm^3. At r 5^-, V 0\ cm^3. At r = (10)/(3)\ cm, this is a candidate for maximum. Corresponding h: h = 5 - (10)/(3) = (15)/(3) - (10)/(3) = (5)/(3)\ cm. Step 6: Compute the second derivative to confirm maximum: d^2Vdr^2 = (1)/(3) (d)/(dr) (10 r - 3 r^2) = (1)/(3) (10 - 6 r). At r = (10)/(3)\ cm: 10 - 6 · (10)/(3) = 10 - 20 = -10 < 0, so it is a local maximum. Step 7: Substitute r = (10)/(3)\ cm and h = (5)/(3)\ cm into the volume formula: V = (1)/(3) ( (10)/(3) )^2 ( (5)/(3) )\ cm^3. First, compute ( (10)/(3) )^2: ( (10)/(3) )^2 = (100)/(9). Then, V = (1)/(3) · (100)/(9) · (5)/(3)\ cm^3. Multiply: (100)/(9) · (5)/(3) = (500)/(27). So, V = (1)/(3) · (500)/(27) = (500 )/(81)\ cm^3. The maximum volume is (500)/(81)\ cm^3.