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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Health An epidemic was spreading such that t weeks after its outbreak it had infected

N 1 ( t ) = 0.1 t 2 + 0.5 t + 150 , 0 t 50

people. Twenty-five weeks after the outbreak, a vaccine was developed and administered to the public. At that point, the number of people infected was governed by the model

N 2 ( t ) = 0.2 t 2 + 6 t + 200.

Approximate the number of people that the vaccine prevented from becoming ill during the epidemic.

To determine

To calculate: The number of people prevented by the vaccine from becoming ill during the epidemic. The number of people infected by epidemic t weeks after its outbreak is modelled by the function N1(t)=0.1t2+0.5t+150 and after the development of vaccine the number of people infected by the epidemic follow the model N2(t)=0.2t2+6t+200.

Explanation

Given Information:

The number of people infected by an epidemic t weeks after its outbreak is modelled by the function

N1(t)=0.1t2+0.5t+150,    0t50

25 weeks after the outbreak, a vaccine was developed. The number of people infected by the epidemic follow the model

N2(t)=0.2t2+6t+200

Formula used:

Area of a region bounded by two graphs is calculated using the following formula,

If f and g are continuous on [a,b] and g(x)f(x) for all x in [a,b], then the area of the region bounded by the graphs of f, g, x=a and x=b is given by

A=ab[f(x)g(x)]dx

Calculation:

The number of people that the vaccine prevented from becoming ill during the epidemic will be represented as the area of the region between the graphs of N1 and N2 in the interval [25,50].

As N2N1 for all t in the interval [25,50], the number of people prevented will be,

Number of people prevented=2550[N1(t)N2(t)]dt=2550[(0

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