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.
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Step 1: Recall the linear relationship The relationship between (1)/(I) and R for a simple circuit with internal resistance is linear and can be expressed as: (1)/(I) = (1)/(E)R + (r)/(E) This is in the form of a straight line equation y = mx + c, where y = (1)/(I), x = R, m = (1)/(E) (slope), and c = (r)/(E) (y-intercept). The intercept of the (1)/(I) axis is the value of c. Step 2: Calculate the slope (m) of the line of best fit We use the given data points (R, (1)/(I)): (2 \, , 5.00 \, A^-1), (4 \, , 5.60 \, A^-1), (6 \, , 10.00 \, A^-1), (7 \, , 12.50 \, A^-1), (9 \, , 16.70 \, A^-1). Let x = R and y = (1)/(I). We calculate the necessary sums for the least squares method: x = 2+4+6+7+9 = 28 y = 5.00+5.60+10.00+12.50+16.70 = 49.80 x^2 = 2^2+4^2+6^2+7^2+9^2 = 4+16+36+49+81 = 186 xy = (2 × 5.00) + (4 × 5.60) + (6 × 10.00) + (7 × 12.50) + (9 × 16.70) xy = 10.00 + 22.40 + 60.00 + 87.50 + 150.30 = 330.20 Number of data points n = 5. The slope m is given by: m = (n xy - ( x)( y))/(n x^2 - ( x)^2) m = (5(330.20) - (28)(49.80))/(5(186) - (28)^2) m = (1651 - 1394.4)/(930 - 784) m = (256.6)/(146) m ≈ 1.7575 \, A^-1^-1 Step 3: Calculate the y-intercept (c) The y-intercept c is given by: c = ( y - m x)/(n) Using the calculated slope m ≈ 1.7575: c = (49.80 - (1.7575)(28))/(5) c = (49.80 - 49.21)/(5) c = (0.59)/(5) c = 0.118 \, A^-1 The intercept of the (1)/(I) axis is 0.118 \, A^-1. That's 2 down. 3 left today — send the next one.