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: Identify the given information. Let the two forces be F_1 and F_2. Let the resultant force be R. Given: • The resultant force R = 18 \, N. • One of the forces, let's call it F_1 = 12 \, N. • The resultant R is at right angles (90^) to the smaller force. Step 2: Apply the condition that the resultant is perpendicular to one of the forces. Let's assume F_1 is the smaller force. This means the angle between the resultant R and F_1 is 90^. When the resultant of two forces is perpendicular to one of the forces, a right-angled triangle is formed by the three forces. The larger of the two original forces (F_2) acts as the hypotenuse, and the resultant (R) and the smaller force (F_1) act as the two legs. Therefore, we can use the Pythagorean theorem: F_2^2 = R^2 + F_1^2 Step 3: Substitute the given values and solve for the unknown force. We have R = 18 \, N and F_1 = 12 \, N. F_2^2 = (18 \, N)^2 + (12 \, N)^2 F_2^2 = 324 \, N^2 + 144 \, N^2 F_2^2 = 468 \, N^2 F_2 = sqrt(468) \, N Step 4: Simplify the result and verify the assumption. To simplify sqrt(468): 468 = 4 × 117 = 4 × 9 × 13 = 36 × 13 F_2 = sqrt(36 × 13) \, N F_2 = 6sqrt(13) \, N Now, let's verify our assumption that F_1 is the smaller force. F_1 = 12 \, N. F_2 = 6sqrt(13) \, N. Since sqrt(13) is approximately 3.606, F_2 ≈ 6 × 3.606 = 21.636 \, N. Since 12 \, N < 21.636 \, N, our assumption that F_1 is the smaller force is correct. The other force is 6sqrt(13) \, N. Last free one today — make it count tomorrow, or type /upgrade for unlimited.