   Chapter 10.1, Problem 58E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Medication Suppose that the concentration C of a medication in the bloodstream t hours after an injection is given by C ( t ) = 0.2 t t 2 + 1 (a) Determine the number of hours before C attains its maximum.(b) Find the maximum concentration.

(a)

To determine

To calculate: The number of hours before C attains its maximum if the function is C(t)=0.2tt2+1.

Explanation

Given Information:

The provided equation is C(t)=0.2tt2+1.

Formula Used:

As per the First Derivative Test,

Step 1: Evaluate the first derivative of the function is evaluated.

Step 2: The first derivative is made equal to zero in order to get the critical points that satisfy f(x)=0. The values which makes first derivative of the function undefined known as critical values.

Step 3: Critical values substitute into the original function to get the critical points.

Step 4: Now find f(x) at a value of x to the left and right of each critical to get either function is maximum or minimum.

If f(x)>0 to the left and f(x)<0 to the right of the critical value, the critical point is a relative maximum point.

If f(x)<0 to the left and f(x)>0 to the right of the critical value, the critical point is a relative minimum point.

Calculation:

Consider the provided equation,

C(t)=0.2tt2+1

The critical values are the only values at which the graph can have turning points, the derivative cannot change sign anywhere except at the critical value.

Hence, there will no change in the values of critical values as in the derivative graph.

Take out the first derivative of the equation by the power rule,

C=(t2+1)(0.2)0.2t(2t)(t2+1)2=0.2t2+0.2(t2+1)2

Put the value of C=0,

0.2t2+0.2(t2+1)2=00.2t2+0.2=0t2=1

Evaluate the value of t from the equation:

t2=1t=±1

As the negative value is not in the domain

(b)

To determine

To calculate: The maximum concentration if the function is C(t)=0.2tt2+1 if suppose that the concentration C of a medication in the bloodstream t hours after an injection is given as above.

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