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Chapter 1: Problem 47

During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate of 2 inches per hour for the next 6 hours, and at a rate of 0.5 inch per hour for the final hour. Write and graph a piecewise- defined function that gives the depth of the snow during the snowstorm. How many inches of snow accumulated from the storm?

Short Answer

Expert verified
The total inches of snow that accumulated from the storm is 14.5 inches.

Step by step solution

01

Writing the Piecewise Function for the Different Snowfall Rates

The piecewise function for the snowfall rate can be described as this:\[s(t) = \begin{cases} 1t & 0 \leq t < 2 \2(t-2) + 2 & 2 \leq t < 8 \0.5(t-8) + 14 & 8 \leq t \leq 9\end{cases}\]The first section of the function covers the period from t=0 to t=2 hours with a 1 inch per hour rate. The second section covers from t=2 to t=8 hours with a rate of 2 inches per hour. The '2' in the second section represents the total snowfall of the first section. The third section covers from t=8 to t=9 hours with a rate of 0.5 inch per hour. The '14' in this section represents the total snowfall from the first and second sections.
02

Calculate the Snowfall for Each Time Interval

Now, calculate the snow depth for each section by multiplying the snow rate by the hours:1. For the first 2 hours, it snows at a rate of 1 inch per hour. So the total snow is \(1 \times 2 = 2\) inches.2. For the next 6 hours, it snows at a rate of 2 inches per hour, so the total snow for this interval is \(2 \times 6 = 12\) inches.3. For the last hour, it snows at a rate of 0.5 inch per hour, hence, the total snow for this interval is \(0.5 \times 1 = 0.5\) inch.
03

Calculate the Total Snow Accumulation

Finally, to find the total snowfall depth, sum the depth from each interval:\[2 + 12 + 0.5 = 14.5 \text{ inches}\]So, 14.5 inches of snow accumulated from the storm.

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