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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The volume of a cylinder is given by the formula V = r^2 h. Step 1: Calculate the initial volume of the cylindrical container. Given radius r = 14 cm and height h = 5 cm. V_initial = (14)^2 (5) = (196)(5) = 980 cm^3 Step 2: Calculate the increase in volume when the radius is increased by x cm. New radius r' = (14+x) cm, height remains h = 5 cm. New volume V_r = (14+x)^2 (5) Increase in volume V_r = V_r - V_initial V_r = 5 (14+x)^2 - 980 V_r = 5 [(14+x)^2 - 196] V_r = 5 [196 + 28x + x^2 - 196] V_r = 5 (x^2 + 28x) Step 3: Calculate the increase in volume when the height is increased by x cm. Radius remains r = 14 cm, new height h' = (5+x) cm. New volume V_h = (14)^2 (5+x) Increase in volume V_h = V_h - V_initial V_h = 196 (5+x) - 980 V_h = 980 + 196 x - 980 V_h = 196 x Step 4: Equate the two increases in volume and solve for x. The problem states that the increase in volume is the same in either case. V_r = V_h 5 (x^2 + 28x) = 196 x Divide both sides by (since ≠ 0): 5(x^2 + 28x) = 196x 5x^2 + 140x = 196x 5x^2 + 140x - 196x = 0 5x^2 - 56x = 0 Factor out x: x(5x - 56) = 0 This gives two possible solutions: x=0 or 5x-56=0. Since the increase in volume is "not zero", x cannot be 0. Therefore, we take the second solution: 5x - 56 = 0 5x = 56 x = (56)/(5) x = 11.2 cm Comparing this with the given options, the correct option is (D). The final answer is D) 11.2 cm. That's 2 down. 3 left today — send the next one.