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
You're on a roll — Here's the solution for question 4. Step 1: Define the initial and final volumes and radii. Let the initial volume of the spherical balloon be V_0 and its initial radius be r_0. Let the final volume be V_1 and its final radius be r_1. The formula for the volume of a sphere is V = (4)/(3) r^3. So, we have: V_0 = (4)/(3) r_0^3 V_1 = (4)/(3) r_1^3 Step 2: Express the new volume (V_1) in terms of the initial volume (V_0) using the given percentage increase. The volume increases by 18p\%. This means the new volume is the original volume plus 18p\% of the original volume. V_1 = V_0 + (18p)/(100)V_0 V_1 = V_0 (1 + (18p)/(100)) Step 3: Substitute the volume formulas from Step 1 into the equation from Step 2. (4)/(3) r_1^3 = (4)/(3) r_0^3 (1 + (18p)/(100)) Step 4: Solve for r_1 in terms of r_0. Divide both sides of the equation by (4)/(3): r_1^3 = r_0^3 (1 + (18p)/(100)) Take the cube root of both sides: r_1 = [3]r_0^3 (1 + (18p)/(100)) r_1 = r_0 (1 + (18p)/(100))^1/3 Step 5: Calculate the percentage change in radius. The percentage change in radius is given by the formula: Percentage change = Change in radiusOriginal radius × 100\% Percentage change = (r_1 - r_0)/(r_0) × 100\% Substitute the expression for r_1 from Step 4: Percentage change = (r_0 (1 + 18p)/(100))^1/3 - r_0r_0 × 100\% Factor out r_0 from the numerator: Percentage change = (r_0 [ (1 + 18p)/(100))^1/3 - 1 ]r_0 × 100\% Percentage change = [ (1 + (18p)/(100))^1/3 - 1 ] × 100\% The percentage change in its radius is [ (1 + (18p)/(100))^1/3 - 1 ] × 100\%. Send me the next one 📸