Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 11.1, Problem 55E

(a)

To determine

To calculate: The length of the path traveled by the projectile.

(a)

Expert Solution

Answer to Problem 55E

The required length of the path traveled by the projectile is 840.421 feet.

Explanation of Solution

Given information:

The parametric equations: $x=(v0cosθ)t$ , $y=(v0sinθ)t−16t2$

The angle: $θ=60°$

The initial velocity: $v0=150$

Calculation:

The given parametric equations are $x=(v0cosθ)t$ and $y=(v0sinθ)t−16t2$

Differentiate the parametric equations with respect to t .

$dxdt=v0cosθ$ , $dydt=v0sinθ−32t$

At t = 0; $y=0$

Substitute 0 for y in the expression $y=(v0sinθ)t−16t2$ .

$0=(v0sinθ)t−16t2t(v0sinθ−16t)=0$

Thus, $t=0$

And, $v0sinθ−16t=016t=v0sinθt=v0sinθt$

Consider that L be the length of the parametric curve.

So, $L=∫t2t2(dxdt)2+(dydt)2dt$

Write the length of the path traveled by the projectile as:

$L=∫0v0sinθ16(v0cosθ)2+(v0sinθ−32t2)dt$

Also,

$dydt=0v0sinθ−32t=0t=v0sinθ32$

The maximum height at $t=v0sinθ32$ is:

$ymax=(v0sinθ)(v0sinθ32)−16(v0sinθ32)2$

Now, substitute $60°$ for $θ$ and 150 for $v0$ in the expression $L=∫0v0sinθ16(v0cosθ)2+(v0sinθ−32t2)dt$ .

$L=∫075sin60°8(150cos60°)2+(150sin60°−32t2)dt=840.421$

Hence, the required length of the path traveled by the projectile is 840.421 feet.

(b)

To determine

To calculate: The maximum height of the projectile.

(b)

Expert Solution

Answer to Problem 55E

The required maximum height of the projectile is 263.672 feet.

Explanation of Solution

Given information:

The parametric equations: $x=(v0cosθ)t$ , $y=(v0sinθ)t−16t2$

The angle: $θ=60°$

The initial velocity: $v0=150$

Calculation:

The given parametric equations are $x=(v0cosθ)t$ and $y=(v0sinθ)t−16t2$

Differentiate the parametric equations with respect to t .

$dxdt=v0cosθ$ , $dydt=v0sinθ−32t$

At t = 0; $y=0$

Substitute 0 for y in the expression $y=(v0sinθ)t−16t2$ .

$0=(v0sinθ)t−16t2t(v0sinθ−16t)=0$

Thus, $t=0$

And, $v0sinθ−16t=016t=v0sinθt=v0sinθt$

Consider that L be the length of the parametric curve.

So, $L=∫t2t2(dxdt)2+(dydt)2dt$

Write the length of the path traveled by the projectile as:

$L=∫0v0sinθ16(v0cosθ)2+(v0sinθ−32t2)dt$

Also,

$dydt=0v0sinθ−32t=0t=v0sinθ32$

The maximum height at $t=v0sinθ32$ is:

$ymax=(v0sinθ)(v0sinθ32)−16(v0sinθ32)2$

Substitute $60°$ for $θ$ and $150$ for $v0$ in the expression $ymax=(v0sinθ)(v0sinθ32)−16(v0sinθ32)2$ .

$ymax=(150sin60°)(150sin60°32)−16(150sin60°32)2=263.672$

Hence, the required maximum height of the projectile is 263.672 feet.

Chapter 11.1, Problem 54E

Chapter 11.1, Problem 56E

Chapter 11 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

Ch. 11.1 - Prob. 1QR Ch. 11.1 - Prob. 2QR Ch. 11.1 - Prob. 3QR Ch. 11.1 - Prob. 4QR Ch. 11.1 - Prob. 5QR Ch. 11.1 - Prob. 6QR Ch. 11.1 - Prob. 7QR Ch. 11.1 - Prob. 8QR Ch. 11.1 - Prob. 9QR Ch. 11.1 - Prob. 10QR

Additional Math Textbook Solutions

Find more solutions based on key concepts

In Exercises 5-36, express all probabilities as fractions. 23. Combination Lock The typical combination lock us...

Elementary Statistics

A categorical variable has three categories, with the following frequencies of occurrence: a. Compute the perce...

Basic Business Statistics, Student Value Edition

Fill in each blanks so that the resulting statement is true. Any set of ordered pairs is called a/an _______. T...

College Algebra (7th Edition)

Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checkli...

Calculus: Early Transcendentals (2nd Edition)

Find the point-slope form of the line passing through the given points. Use the first point as (x1, .y1). Plot ...

College Algebra with Modeling & Visualization (5th Edition)

In Exercises 13–16, find the margin of error for the values of c, ?, and n. 16. e = 0.975, ? = 4.6, n = 100

Elementary Statistics: Picturing the World (7th Edition)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.