SNOWSTORM During a nine-hour snowstorm, it snows at a rate...

Name: Problem 72, Chapter 1: Precalculus with Limits
Uploaded: 2022-05-28T01:19:08-08:00
Duration: 4 min 17 s
Channel: Mary Wakumoto
Description: SNOWSTORM 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?

SNOWSTORM 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?

Key Concepts

Graphing Functions

Graphing a function involves representing its behavior visually on a coordinate plane, which is particularly beneficial for piecewise-defined functions as it clearly shows how the function changes over its different intervals. This visualization aids in understanding discontinuities, slopes, and overall trends in the modeled behavior.

Rate of Change and Accumulation

Rate of change describes how a quantity varies over time, and accumulation involves summing these changes over an interval to find total effect. In contexts such as snowfall, the rate at which the snow falls can vary over time, so integrating these rates, whether through summation or definite integration, yields the total accumulation.

Piecewise-Defined Functions

A piecewise-defined function is a function that is defined by different expressions over various intervals of its domain. This approach is useful in modeling scenarios where the behavior of a quantity changes at different stages. Understanding how to construct and interpret such functions allows one to analyze and compute values based on the input's location in the domain.

Transcript

00:02 All right, so here we have a snowstorm.

00:05 And the first two hours, we have one inch coming down per hour.

00:08 That's the set in black.

00:10 Then in red, we have two inches coming down per hour for those six hours between two and eight hours.

00:17 And finally, just a half inch per hour at the last hour.

00:21 And our goal is to come up with the accumulation totals and basically with time as a piecewise function.

00:28 So let's solve each of these scenarios.

00:31 Okay, so if df dt equals 1, then if i cross multiply, df equals dt.

00:38 So i'm going to integrate both sides and that will give me f equals t plus c.

00:44 We can solve for c because we know when t is zero, we don't have any snowfall.

00:50 So zero equals zero plus c, c equals zero.

00:54 So the first part of my function is just going to be f a t.

01:01 Equals t.

01:03 So down here i'm going to make the piecewise function keep track of everything as we go along.

01:09 Okay, so f a t for the first two hours, so zero less than t less than equal to two is just t.

01:19 All right, now let's go ahead and look at the heavy snow, the two inches per hour between two and eight.

01:27 I'm going to do the same thing.

01:28 I'm going to separate variables.

01:30 This time i get two dt, i'm the right integrate both sides f then is two t plus c well let's see where we're at at two hours if i plug in f a two to my first set then i get two inches so at the end of when the first part first section's done i'm at two inches so i can say that f at two equals twice t plus c solve for c and that has to equal to inches.

02:02 So therefore, c must be we have four, so c is two minus four or minus two.

02:10 So c then is minus two.

02:12 So f of t then is two t minus two.

02:17 So that goes here, two t minus two, and that's in between two and eight, eight hours.

02:26 Okay, last set, but before we do it, we are...

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