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Chapter 8: Problem 35

A climbing plane has vertical velocity \(v(h)\) meters/second, where \(h\) is its altitude in meters. Write a definite integral for the time it takes the plane to climb 7000 meters from the ground.

Short Answer

Expert verified
\( t = \int_{0}^{7000} \frac{1}{v(h)} \, dh \) is the integral for the climbing time.

Step by step solution

01

Understand the Problem

We are given the vertical velocity of a climbing plane as a function of altitude, denoted by \( v(h) \). Our task is to write a definite integral that represents the time it takes for the plane to climb from 0 to 7000 meters.
02

Identify the Integral Representation

The time \( t \) taken to travel a certain vertical distance with varying velocity is given by the integral of the reciprocal of the velocity function. This can be expressed as \( t = \int_{0}^{7000} \frac{1}{v(h)} \, dh \). Here, \( \frac{1}{v(h)} \) gives the differential time per unit altitude.
03

Set Up the Definite Integral

Given the information, we set up the definite integral for the total climbing time as a definite integral from 0 to 7000 meters: \[ t = \int_{0}^{7000} \frac{1}{v(h)} \, dh \] This integral represents the total time the plane will take to climb 7000 meters, assuming \( v(h) \) is known and integrable.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Velocity
Vertical velocity is a crucial concept when understanding how a plane climbs. It refers to how fast the plane moves upwards through the sky, measured in meters per second (m/s). Imagine it as the speed at which the plane's altitude increases.
In the case of our climbing plane, this vertical velocity varies with altitude. So, as the altitude changes, the speed at which the plane ascends also changes. This is denoted as the function \( v(h) \), where \( h \) is the altitude.
We need to pay close attention to how this speed changes as the plane climbs, because that influences how long it will take to reach a certain altitude. This variation is key to understanding the problem.
Altitude
Altitude is simply the height above the ground level, typically measured in meters or feet. For a climbing plane, altitude is what the plane aims to increase when ascending.
It's important to understand that an increase in altitude can affect vertical velocity. In this exercise, altitude starts at 0 meters (ground level) and ends at 7000 meters.
Often, certain altitudes can have different atmospheric conditions affecting the plane's velocity. Thus, considering altitude in relation to vertical velocity helps when determining how long the plane will take to climb to its desired height.
Reciprocal Function
A reciprocal function takes a number and flips it. The reciprocal of a function like vertical velocity, \( v(h) \), is represented as \( \frac{1}{v(h)} \).
For us, this reciprocal tells how much time it takes to climb a small increment of altitude; it's the differential time per unit altitude. This is because time is the reciprocal of speed from a mathematical perspective.
Using the reciprocal helps us translate velocity into time taken to climb a given altitude. This is the core of setting up the definite integral used in solving the problem.
Climbing Time
Climbing time is the total time a plane takes to ascend from one altitude to another. In this context, it is the time taken to climb from 0 to 7000 meters.
We use definite integrals to calculate climbing time when vertical velocity varies with each altitude. The key to finding this time is integrating the reciprocal of the velocity function over the altitude range.
Our integral \( t = \int_{0}^{7000} \frac{1}{v(h)} \, dh \) computes the total climbing time, taking into account how velocity varies with altitude. By solving this integral, we get a clear picture of the entire climbing process in terms of time.

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