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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.8.43c
Chapter 8, Problem 8.8.43c

43. A hot-air balloon is launched from an elevation of 5400 ft above sea level. As it rises, the vertical velocity is computed using a device (called a variometer) that measures the change in atmospheric pressure. The vertical velocities at selected times are shown in the table (with units of ft/min).
tab1
c. A polynomial that fits the data reasonably well is:
g(t) = 3.49t³ - 43.21t² + 142.43t - 1.75
Estimate the elevation of the balloon after five minutes using this polynomial.

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Step 1: Understand the problem. We are tasked with estimating the elevation of the hot-air balloon after 5 minutes using the given polynomial g(t) = 3.49t³ - 43.21t² + 142.43t - 1.75. The initial elevation of the balloon is 5400 ft above sea level.
Step 2: Recognize that the polynomial g(t) represents the vertical velocity of the balloon as a function of time t (in minutes). To find the elevation, we need to integrate g(t) with respect to t, as the elevation is the accumulation of vertical velocity over time.
Step 3: Set up the integral for the elevation. The elevation E(t) at time t is given by the integral of g(t) from 0 to t, plus the initial elevation. Mathematically, this is expressed as: E(t) = 5400 + ∫[0 to t] g(t) dt.
Step 4: Compute the indefinite integral of g(t). The integral of g(t) = 3.49t³ - 43.21t² + 142.43t - 1.75 is: ∫g(t) dt = (3.49/4)t⁴ - (43.21/3)t³ + (142.43/2)t² - 1.75t + C, where C is the constant of integration.
Step 5: Evaluate the definite integral from 0 to 5. Substitute t = 5 into the integrated polynomial and subtract the value of the polynomial at t = 0. Add the result to the initial elevation of 5400 ft to find the estimated elevation after 5 minutes.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this context, the polynomial g(t) = 3.49t³ - 43.21t² + 142.43t - 1.75 models the vertical velocity of the hot-air balloon over time. Understanding how to evaluate polynomial functions is crucial for estimating values, such as the elevation of the balloon after a specific time.
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Integration

Integration is a fundamental concept in calculus that allows us to find the accumulated value of a function over an interval. In this scenario, to estimate the elevation of the balloon after five minutes, we need to integrate the velocity function g(t) over the interval from 0 to 5 minutes. This process gives us the total change in elevation, which we then add to the initial elevation of 5400 ft.
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Initial Conditions

Initial conditions refer to the starting values of a function at a specific point, which are essential for solving differential equations or evaluating functions. In this problem, the initial elevation of the hot-air balloon is 5400 ft above sea level. This value is critical as it serves as the baseline from which we calculate the balloon's elevation after applying the results from the integration of the velocity function.
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