Question

number of dogs time spent walking (minutes) 1 25 2 36 3 40 4 52 5 50 6 68 7 74 8 110 9 89 10 92 question 5 if kai walked 12 dogs, what might be a reasonable estimate for the walking time? justify your reasoning.

Explanation:

Step1: Find the trend

Looking at the data, as the number of dogs increases, the time generally increases.

Step2: Estimate the rate

From \(1\) dog (\(25\) minutes) to \(2\) dogs (\(36\) minutes), the increase is \(36 - 25=11\) minutes. From \(2\) dogs to \(3\) dogs (\(40\) minutes), the increase is \(40 - 36 = 4\) minutes. From \(3\) dogs to \(4\) dogs (\(52\) minutes), the increase is \(52-40 = 12\) minutes. From \(4\) dogs to \(5\) dogs (\(50\) minutes), there is a decrease of \(52 - 50=2\) minutes. From \(5\) dogs to \(6\) dogs (\(68\) minutes), the increase is \(68 - 50=18\) minutes. From \(6\) dogs to \(7\) dogs (\(74\) minutes), the increase is \(74 - 68 = 6\) minutes. From \(7\) dogs to \(8\) dogs (\(110\) minutes), the increase is \(110 - 74=36\) minutes. From \(8\) dogs to \(9\) dogs (\(89\) minutes), there is a decrease of \(110 - 89 = 21\) minutes. From \(9\) dogs to \(10\) dogs (\(92\) minutes), the increase is \(92 - 89=3\) minutes.

If we consider a rough linear trend (ignoring some fluctuations), from \(10\) dogs (\(92\) minutes), if we assume an average increase of about \(10\) minutes per \(2\) dogs (a very rough estimate based on some of the larger intervals).

For \(12\) dogs ( \(2\) more than \(10\) dogs), a reasonable estimate could be \(92+ 20=112\) minutes. Another way: If we consider the data points and try to fit a line of best - fit conceptually (without formal regression). The values seem to have an upward trend. If we look at the higher - end values (e.g., when the number of dogs is \(8\) (\(110\) minutes), \(9\) (\(89\) minutes), \(10\) (\(92\) minutes)), and assume that for \(12\) dogs, since the trend is still upwards (despite some fluctuations), a value around \(110 - 130\) minutes could be reasonable. If we consider that from \(10\) dogs to \(12\) dogs, and if we assume a more conservative increase (using the average of some of the smaller increases). The average of \(11,4,12,6,3\) (some of the non - extreme increases) is \(\frac{11 + 4+12+6+3}{5}=\frac{36}{5} = 7.2\). For \(2\) more dogs (from \(10\) to \(12\)), an increase of about \(14\) minutes. So \(92+14 = 106\) minutes. A middle - of - the - range estimate considering the fluctuations could be around \(115\) minutes.

Answer:

A reasonable estimate for walking \(12\) dogs could be \(115\) minutes. The reasoning is based on observing the upward trend in the data (despite some fluctuations) and making a rough linear - like extrapolation. When looking at the values for \(8\) (\(110\) minutes), \(9\) (\(89\) minutes), \(10\) (\(92\) minutes), and assuming that the trend continues to rise for \(12\) dogs, \(115\) minutes is a value that fits within the range of possible upward extensions of the data pattern.