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
Let's check your work for the standard deviation and then complete the circle problem. Standard Deviation Problem The scores are 10, 12, 14, 16, 28, 30. The assumed mean A = 15. The deviations d_i = x_i - A are: 10-15 = -5 12-15 = -3 14-15 = -1 16-15 = 1 28-15 = 13 30-15 = 15 Step 1: Calculate the sum of deviations d_i. d_i = -5 + (-3) + (-1) + 1 + 13 + 15 = 20 Your calculation (20)/(6) = 3.33 is correct for ( d_i)/(n), which is a correction factor for the mean, not the mean itself. The actual mean x = A + ( d_i)/(n) = 15 + 3.33 = 18.33. However, this does not affect the standard deviation calculation using the assumed mean method. Step 2: Calculate the sum of squared deviations d_i^2. d_i^2 = (-5)^2 + (-3)^2 + (-1)^2 + (1)^2 + (13)^2 + (15)^2 d_i^2 = 25 + 9 + 1 + 1 + 169 + 225 = 430 Your calculation (430)/(6) = 71.67 is correct for ( d_i^2)/(n). Step 3: Calculate the variance V using the formula V = ( d_i^2)/(n) - (( d_i)/(n))^2. V = 71.67 - (3.33)^2 V = 71.67 - 11.0889 V = 60.5811 Rounding to two decimal places, V ≈ 60.58. Your calculation is correct. Step 4: Calculate the standard deviation = sqrt(V). = sqrt(60.58) ≈ 7.7833 Rounding to two decimal places, ≈ 7.78. Your calculation is correct. The standard deviation is 7.78. --- 9. The equation of a circle The equation of the circle is given as x^2+y^2-4x+6y+4=0. Step 1: Group the x terms and y terms, and move the constant to the right side. (x^2-4x) + (y^2+6y) = -4 Step 2: Complete the square for the x terms and y terms. To complete the square for x^2-4x, add ((-4)/(2))^2 = (-2)^2 = 4. To complete the square for y^2+6y, add ((6)/(2))^2 = (3)^2 = 9. Add these values to both sides of the equation. (x^2-4x+4) + (y^2+6y+9) = -4 + 4 + 9 Step 3: Rewrite the expressions in squared form. (x-2)^2 + (y+3)^2 = 9 Your work matches this step perfectly. Step 4: Identify the center and radius of the circle. The standard form of a circle's equation is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius. Comparing this to (x-2)^2 + (y+3)^2 = 9: The center of the circle is (h,k) = (2, -3). The radius of the circle is r = sqrt(9) = 3. Step 5: Describe how to draw the circle on a grid. To draw the circle: • Plot the center point at (2, -3) on the grid. • From the center, measure 3 units in all four cardinal directions (up, down, left, right) to mark points (2, 0), (2, -6), (-1, -3), and (5, -3). • Sketch a smooth curve connecting these points to form the circle. The center of the circle is (2, -3) and the radius is 3. What's next? Send 'em!