Question

There was no snow on the ground when it started falling at midnight at a constant rate of 1.5 inches per hour. At

4:00 a.m., it starting falling at a constant rate of 3 inches per hour, and then from 7:00 a.m. to 9:00 a.m., snow was
falling at a constant rate of 2 inches per hour. It stopped snowing at 9:00 a.m. (Note: This problem models snow
falling by a constant rate during each time period. In reality, the snowfall rate might be very close to constant but is
unlikely to be perfectly uniform throughout any given time period.)
c. When was the depth of the snow on the ground 8 inches?

Answer 1

`x=(14)/(3)` hours after midnight or at 4:40 am the depth of the snow on the ground was 8 inches.

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`

At 4:00 a.m., it starting falling at a constant rate of 3 inches per hour,

`S(x)=3(x-4)+6\ \ \ 4\leq x<7`

7:00 a.m. to 9:00 a.m., snow was falling at a constant rate of 2 inches per hour.

`S(x)=2(x-7)+15\ \ \ 7\leq x\leq 9`

The required piece-wise linear function is

`\left\{\begin{matrix}1.5x & 0\leq x<4\\ 3(x-4)+6 & 4\leq x<7\\ 2(x-7)+15 & 7\leq x\leq 9\end{matrix}\right.`

Now we need to find When was the depth of the snow on the ground 8 inches?

Substitute S(x)=8 in
`S(x)=3(x-4)+6`

`8=3(x-4)+6`

`2=3(x-4)`

`(2)/(3)=x-4`

`(2+12)/(3)=x`

`x=(14)/(3)`

`x=(14)/(3)` hours after midnight or at 4:40 am the depth of the snow on the ground was 8 inches.