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
To estimate the values of a and b from the given expression y = ax^2 + bx, we first need to transform it into the equation of a straight line, Y = mX + c. Step 1: Rearrange the equation into a linear form. The given equation is y = ax^2 + bx. To make it linear, we can divide the entire equation by x (assuming x ≠ 0): (y)/(x) = (ax^2)/(x) + (bx)/(x) (y)/(x) = ax + b Now, let Y = (y)/(x) and X = x. The equation becomes: Y = aX + b This is in the form of a straight line Y = mX + c, where the gradient m = a and the Y-intercept c = b. Step 2: Calculate the values for X and Y using the given table. The original table is: | x | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---| | y | 6 | 16 | 30 | 48 | 70 | 96 | Now, we calculate Y = (y)/(x) for each pair: • For x=1, y=6: Y = (6)/(1) = 6 • For x=2, y=16: Y = (16)/(2) = 8 • For x=3, y=30: Y = (30)/(3) = 10 • For x=4, y=48: Y = (48)/(4) = 12 • For x=5, y=70: Y = (70)/(5) = 14 • For x=6, y=96: Y = (96)/(6) = 16 The new table for plotting the straight-line graph is: | X = x | Y = (y)/(x) | |---------|-------------------| | 1 | 6 | | 2 | 8 | | 3 | 10 | | 4 | 12 | | 5 | 14 | | 6 | 16 | Step 3: Draw a suitable straight-line graph. To draw the graph: • Plot the points (X, Y) from the table above on a graph paper. • The horizontal axis (X-axis) should represent x. • The vertical axis (Y-axis) should represent (y)/(x). • Draw a straight line that passes through all these points. (In this case, the points are perfectly collinear). Step 4: Estimate the values of a and b from the graph. From the linear equation Y = aX + b: • The value of a is the gradient of the straight line. • The value of b is the Y-intercept (the value of Y when X=0). To find the gradient a: We can use any two points from the transformed table, for example, (X_1, Y_1) = (1, 6) and (X_2, Y_2) = (6, 16). a = (Y_2 - Y_1)/(X_2 - X_1) = (16 - 6)/(6 - 1) = (10)/(5) = 2 To find the Y-intercept b: We can substitute the value of a and any point (X, Y) into the equation Y = aX + b. Let's use the point (1, 6): 6 = (2)(1) + b 6 = 2 + b b = 6 - 2 b = 4 Thus, the estimated values are a=2 and b=4. The final answer is a=2, b=4.