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.
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Problem 12: Spherical balloon Step 1: Volume of a sphere. V = (4)/(3) r^3 Step 2: Differentiate both sides with respect to time t (chain rule). (dV)/(dt) = 4 r^2 (dr)/(dt) Step 3: Substitute r = 5 cm and (dr)/(dt) = 2 cm/s. (dV)/(dt) = 4 (5)^2 (2) = 4 (25)(2) = 200 (dV)/(dt) = 200 cm³/s 200 Problem 13: Ladder problem Step 1: Pythagorean theorem for ladder length l = 10 m (constant). x^2 + y^2 = 10^2 = 100 Step 2: Differentiate both sides with respect to time t. 2x (dx)/(dt) + 2y (dy)/(dt) = 0 Step 3: Solve for (dy)/(dt). (dy)/(dt) = -(x)/(y) (dx)/(dt) Step 4: Find y when x = 6 m. y = sqrt(100 - 6^2) = sqrt(100 - 36) = sqrt(64) = 8 m Step 5: Substitute x = 6 m, y = 8 m, (dx)/(dt) = 1 m/s. (dy)/(dt) = -(6)/(8) (1) = -(3)/(4) = -0.75 m/s The top is sliding down at 0.75 m/s. (dy)/(dt) = -(3)/(4) m/s -(3)/(4)