Advanced Mathematical Concepts: Precalculus with Applications, Student Edition

1st Edition

ISBN: 9780078682278

Author: McGraw-Hill, Berchie Holliday

Publisher: Glencoe/McGraw-Hill

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Question

Chapter 5.1, Problem 67E

(a)

To determine

To write: An equation to model the percent of the motor vehicles produced in the United States as a function of the number of years since 1950.

(a)

Expert Solution

Answer to Problem 67E

P=−1.074t+75.7

Explanation of Solution

Given information: Manufacturing: The percent of the motor vehicles produced in the United States since 1950 is depicted in the table given below.

Advanced Mathematical Concepts: Precalculus with Applications, Student Edition, Chapter 5.1, Problem 67E , additional homework tip 1

Calculation:

We need to find y=mx+b . Pick two years to use as point to find the slope. I picked 1950 as year 0, that way any year past that is just a one or two digit number.

So 1950 as a (x,y) pair becomes (0,75.7) .

I also picked the year 2000, so it is 50 years since 1950 and it’s (x,y) becomes (50,22) .

The year 1950 (0,75.7) and the year 2000 (50,22)

Find the slope

y2−y1x2−x1=22−75.750−0=−53.750=−1.074 This is our slope “ m ”.

We know the y intercept.

At x=0(1950)y=75.7

So our equation is

y=−1.074x+75.7

We had used a point to find b . we might get a slightly different answer.

Let’s us the (50,22) point we used in slope equation. y=mx+b becomes

22=−1.074×(50)+b which is 22=−53.7+b and solving for b gives us

b=75.7,

So it worked out.

It may not work out for all points we could have chosen.

Since this is a real life description, it is unlikely the decrease in percent of motor vehicles went down the exact amount each decade. Notice the end of the chart switches to single years after 2000. If we had used one of those points we might really see some problems.

It is not a proper chart (or graph) that changes the intervals in the data in the middle of the chart.

Let’s change the x and y to reflect the reality of our problem. Replace y with P for percent of motor vehicles produced and x with t for the number of years since 1950.

P=−1.074t+75.7

(b)

To determine

To find:percent of motor vehicles will be produced in the United States in the year 2010, according to the equation.

(b)

Expert Solution

Answer to Problem 67E

11.26 percent

Explanation of Solution

Given information: Manufacturing: The percent of the motor vehicles produced in the United States since 1950 is depicted in the table given below.

Advanced Mathematical Concepts: Precalculus with Applications, Student Edition, Chapter 5.1, Problem 67E , additional homework tip 2

Calculation:

P=−1.074t+75.7

We want to know the percent in 2010 which means t = 60

P=−1.074×60+75.7=−64.44+75.7=11.26

Hence,

11.26 percent

Chapter 5.1, Problem 66E

Chapter 5.1, Problem 68E

Chapter 5 Solutions

Advanced Mathematical Concepts: Precalculus with Applications, Student Edition

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