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: Define variables and set up equations. Let the length of the rectangular garden be l meters and the width be w meters. The perimeter of a rectangle is P = 2(l+w). Half the perimeter is given as 36 m, so: (2(l+w))/(2) = 36 l+w = 36 (*1) The length is 4 m more than its width: l = w+4 (*2) Step 2: Solve the system of equations. Substitute equation (2) into equation (1): (w+4) + w = 36 2w + 4 = 36 Subtract 4 from both sides: 2w = 36 - 4 2w = 32 Divide by 2: w = (32)/(2) w = 16 m Now substitute the value of w back into equation (*2) to find l: l = 16 + 4 l = 20 m The dimensions of the garden are length = 20 m and width = 16 m. Step 3: Compare with the given options. The calculated dimensions are 20 m, 16 m. This matches option (3). The final answer is 20 m, 16 m. --- Step 1: Write down the given equations. The two linear equations are: 2x+3y-4=0 2x+3y=4 (*1) 3x+4y+1=0 3x+4y=-1 (*2) Step 2: Use the elimination method to solve for x and y. Multiply equation (1) by 3 and equation (2) by 2 to make the coefficients of x equal: 3(2x+3y) = 3(4) 6x+9y=12 (*3) 2(3x+4y) = 2(-1) 6x+8y=-2 (*4) Subtract equation (4) from equation (3): (6x+9y) - (6x+8y) = 12 - (-2) 6x+9y-6x-8y = 12+2 y = 14 Substitute y=14 into equation (*1): 2x+3(14) = 4 2x+42 = 4 Subtract 42 from both sides: 2x = 4-42 2x = -38 Divide by 2: x = (-38)/(2) x = -19 Step 3: Compare with the given options. The solution is x=-19, y=14. This matches option (4). The final answer is x=-19, y=14. --- Step 1: Identify the condition for real and distinct roots. For a quadratic equation in the form ax^2+bx+c=0, the roots are real and distinct if the discriminant = b^2-4ac is greater than 0. Step 2: Apply the condition to the given equation. The given equation is x^2+4x+k=0. Here, a=1, b=4, and c=k. Set the discriminant greater than 0: b^2-4ac > 0 (4)^2 - 4(1)(k) > 0 16 - 4k > 0 Step 3: Solve the inequality for k. Add 4k to both sides: 16 > 4k Divide by 4: (16)/(4) > k 4 > k This can also be written as k < 4. Step 4: Compare with the given options. The condition for real and distinct roots is k < 4. Based on the typical structure of such multiple-choice questions, option (3) would represent k < 4. The final answer is k < 4. Send me the next one 📸