A. Find the equation of the line of best fit for the given table. (Round to the nearest tenths place.)
B. Use the equation to estimate the stopping distance at 90 mi/h.
speed mi/h 10,15,20,30,35,40,45
stopping distance ft 27,44,63,85,109,136,164,196
To find the equation of the line of best fit, we can use linear regression. The table can be represented as (x, y) pairs as follows:
(10, 27), (15, 44), (20, 63), (30, 85), (35, 109), (40, 136), (45, 164), (90, ?)
The first step is to find the slope, denoted by m, which can be calculated using the formula:
m = (n∑xy - ∑x∑y) / (n∑x^2 - (∑x)^2)
where n is the number of data points. In this case, n = 7.
∑xy = (10*27) + (15*44) + (20*63) + (30*85) + (35*109) + (40*136) + (45*164) = 13979
∑x = 10 + 15 + 20 + 30 + 35 + 40 + 45 = 195
∑y = 27 + 44 + 63 + 85 + 109 + 136 + 164 = 628
∑x^2 = (10^2) + (15^2) + (20^2) + (30^2) + (35^2) + (40^2) + (45^2) = 7745
Plugging these values into the slope formula:
m = (7*13979 - (195*628)) / (7*7745 - (195)^2)
m = -1.254
Next, we find the y-intercept, denoted by b, using the formula:
b = (∑y - m∑x) / n
Plugging in the values:
b = (628 - (-1.254*195)) / 7
b = 68.3
Therefore, the equation of the line of best fit, in the form y = mx + b, is:
y = -1.254x + 68.3
To estimate the stopping distance at 90 mi/h, we substitute x = 90 into the equation:
y = (-1.254*90) + 68.3
y = -112.9 + 68.3
y ≈ -44.6
Therefore, the estimated stopping distance at 90 mi/h is approximately -44.6 feet.