Answer:
The required piece-wise linear function is
![\left\{\begin{matrix}1.5x & 0\leq x](https://tex.z-dn.net/?f=%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D1.5x%20%26%200%5Cleq%20x%3C4%5C%5C%203%28x-4%29%2B6%20%26%204%5Cleq%20x%3C7%5C%5C%202%28x-7%29%2B15%20%26%207%5Cleq%20x%5Cleq%209%5Cend%7Bmatrix%7D%5Cright.)
Step-by-step explanation:
Consider the provided information.
Let x represents the number of hours and S(x) represents the depth of snow.
There was no snow on the ground when it started falling at midnight at a constant rate of 1.5 inches per.
That means the depth of snow will be:
![S(x)=1.5x\ \ \ 0\leq x< 4](https://tex.z-dn.net/?f=S%28x%29%3D1.5x%5C%20%5C%20%5C%200%5Cleq%20x%3C%204)
At 4:00 a.m., it starting falling at a constant rate of 3 inches per hour,
From mid night to 4:00 a.m the depth of snow will be 1.5×4=6 inches.
If we want to calculate the total depth of snow from midnight to 7:00 am we need to add 6 inches in 3(x-4) where 4≤x<7
Therefore,
![S(x)=3(x-4)+6\ \ \ 4\leq x](https://tex.z-dn.net/?f=S%28x%29%3D3%28x-4%29%2B6%5C%20%5C%20%5C%204%5Cleq%20x%3C7%20)
7:00 a.m. to 9:00 a.m., snow was falling at a constant rate of 2 inches per hour.
From mid night to 7:00 a.m the depth of snow will be 3(7-4)+6=15 inches.
If we want to calculate the total depth of snow from midnight to 9:00 am we need to add 15 inches in 2(x-7) where 7≤x≤9
![S(x)=2(x-7)+15\ \ \ 7\leq x\leq 9](https://tex.z-dn.net/?f=S%28x%29%3D2%28x-7%29%2B15%5C%20%5C%20%5C%207%5Cleq%20x%5Cleq%209%20)
Therefore, the required piece-wise linear function is
![\left\{\begin{matrix}1.5x & 0\leq x](https://tex.z-dn.net/?f=%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D1.5x%20%26%200%5Cleq%20x%3C4%5C%5C%203%28x-4%29%2B6%20%26%204%5Cleq%20x%3C7%5C%5C%202%28x-7%29%2B15%20%26%207%5Cleq%20x%5Cleq%209%5Cend%7Bmatrix%7D%5Cright.)