Question

for a science project, nicholas wants to see if a larger body of water has more heat energy than a smaller body of water at the same temperature. he prepared a number of buckets filled with various amounts of water at a fixed temperature and dropped an ice cube of the same size into each one.
nicholas then recorded the volume of water in each bucket (in milliliters), x, and the amount of time it took for each ice cube to melt (in minutes), y.
round your answers to the nearest thousandth.
y = x +

Explanation:

Step1: Calculate the mean of \(x\) and \(y\)

Let \(x_1 = 149,x_2=323,x_3 = 361,x_4=555,x_5 = 706,x_6=741,x_7 = 840\)
Let \(y_1 = 21.14,y_2=24.91,y_3 = 28.62,y_4=14.21,y_5 = 14.32,y_6=14.18,y_7 = 8.61\)
\(\bar{x}=\frac{149 + 323+361+555+706+741+840}{7}=\frac{3675}{7}=525\)
\(\bar{y}=\frac{21.14 + 24.91+28.62+14.21+14.32+14.18+8.61}{7}=\frac{126}{7}=18\)

Step2: Calculate the numerator and denominator for the slope \(b\)

Numerator: \(\sum_{i = 1}^{7}(x_i-\bar{x})(y_i-\bar{y})\)
\((149 - 525)(21.14 - 18)+(323-525)(24.91 - 18)+(361 - 525)(28.62-18)+(555 - 525)(14.21 - 18)+(706 - 525)(14.32 - 18)+(741-525)(14.18 - 18)+(840 - 525)(8.61 - 18)\)
\(=(-376)\times3.14+(-202)\times6.91+(-164)\times10.62 + 30\times(-3.79)+181\times(-3.68)+216\times(-3.82)+315\times(-9.39)\)
\(=-1180.64-1395.82-1741.68-113.7-666.08-825.12 - 2957.85=-8880.89\)
Denominator: \(\sum_{i = 1}^{7}(x_i-\bar{x})^2\)
\((149 - 525)^2+(323 - 525)^2+(361 - 525)^2+(555 - 525)^2+(706 - 525)^2+(741 - 525)^2+(840 - 525)^2\)
\(=(-376)^2+(-202)^2+(-164)^2+30^2+181^2+216^2+315^2\)
\(=141376+40804+26896 + 900+32761+46656+99225=388618\)
Slope \(b=\frac{-8880.89}{388618}\approx - 0.023\)

Step3: Calculate the intercept \(a\)

Using the formula \(a=\bar{y}-b\bar{x}\), substitute \(b=-0.023\) and \(\bar{x} = 525,\bar{y}=18\)
\(a=18-(-0.023)\times525=18 + 12.075=30.075\)

Answer:

\(y=-0.023x + 30.075\)