Chapter 4.7, Problem 55E

a.

To determine

Write and graph a piecewise function that represents the depth of the snow during snowstorm.

Answer to Problem 55E

  $\begin{array}{l}y=x,\text{if}0<x\le 2\text{and}\\ y=2x-2\text{},\text{}\text{}\text{if}2<x\le 8\text{and}\\ y=x+6,\text{if}8<x\le 9.\end{array}$

Explanation of Solution

Given information:During a 9 hours snowstorm, it snows at a rate of 1 inches per hour for the first 2 hours,2 inches per hour for the next 6 hours, and 1 inches per hour for the final hour.

Calculation:

From the given information the piecewise function is,

  $\begin{array}{l}y=x,\text{if}0<x\le 2\text{and}\\ y=2+2(x-2)\text{},\text{}\text{}\text{if}2<x\le 8\text{and}\\ y=14+1(x-8),\text{if}8<x\le 9.\\ \text{So,}\\ y=x,\text{if}0<x\le 2\text{and}\\ y=2x-2\text{},\text{}\text{}\text{if}2<x\le 8\text{and}\\ y=x+6,\text{if}8<x\le 9.\end{array}$

The first equation is just x since it snows 1 inches per hour for 2 hours. After 2 hours it has snowed 2 inches so the second equation is $y=2+2(x-2)\text{},\text{}\text{}\text{if}2<x\le 8$ and at the 8 hour it has snowed $y=2+2(8-2)=14$ inches .So the last equation is , $y=14+1(x-8),\text{if}8<x\le 9.$

Graph of the piecewise function is given below.

  

b.

To determine

Your friend says 12 inches of snow accumulated during the storm .Is your friend is correct.

Answer to Problem 55E

No, your friend is incorrect . From the graph the point $x=9$ is (9,15) which means 15 inches of snow accumulated over 9 hours.

Explanation of Solution

Given information:During a 9 hours snowstorm, it snows at a rate of 1 inches per hour for the first 2 hours,2 inches per hour for the next 6 hours, and 1 inches per hour for the final hour.

Calculation:

From the given information the piecewise function is,

  $\begin{array}{l}y=x,\text{if}0<x\le 2\text{and}\\ y=2+2(x-2)\text{},\text{}\text{}\text{if}2<x\le 8\text{and}\\ y=14+1(x-8),\text{if}8<x\le 9.\\ \text{So,}\\ y=x,\text{if}0<x\le 2\text{and}\\ y=2x-2\text{},\text{}\text{}\text{if}2<x\le 8\text{and}\\ y=x+6,\text{if}8<x\le 9.\end{array}$

The first equation is just x since it snows 1 inches per hour for 2 hours. After 2 hours it has snowed 2 inches so the second equation is $y=2+2(x-2)\text{},\text{}\text{}\text{if}2<x\le 8$ and at the 8 hour it has snowed $y=2+2(8-2)=14$ inches .So the last equation is , $y=14+1(x-8),\text{if}8<x\le 9.$

Graph of the piecewise function is given below.

  

No, your friend is incorrect . From the graph the point $x=9$ is (9,15) which means 15 inches of snow accumulated over 9 hours.