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Mathematical Applications for the ...

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

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BuyFindarrow_forward

Mathematical Applications for the ...

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

Drug levels in the blood The manufacturer of a medicine wants to test how a new 300-mg capsule is released into the bloodstream. After a volunteer is given a capsule, blood samples are drawn every half hour, and the number of milligrams of the drug in the bloodstream is calculated. The results obtained are shown in the table.

Time t (hr) N(t) (mg) Time t (hr) N(t) (mg)
0 0 2.0 178.3
0.5 247.3 2.5 113.9
1.0 270.0 3.0 56.2
1.5 236.4 3.5 19.3

Use 7 rectangles, each with height N(t) at the left endpoint and with width 0.5 hr, to estimate the area under the graph representing these data. Divide this area by 3.5 hr to estimate the average drug level over this time period.

To determine

To calculate: The area under the graph using 7 rectangles each with the height N(t) measured at the left-handend points and with width 0.5 hr and divide the area by 3.5 hr to estimate the average drug level over this time period where the table represents the number of milligrams of the drug in the blood stream at 0.5 hr time periods.

Time t (hr) N(t)(mg)
0 0
0.5 247.3
1 270
1.5 236.4
2 178.3
2.5 113.9
3 56.2
3.5 19.3
Explanation

Given Information:

The provided table is,

Time t (hr) N(t)(mg)
0 0
0.5 247.3
1 270
1.5 236.4
2 178.3
2.5 113.9
3 56.2
3.5 19.3

Formula used:

The base of the rectangles to approximate the area is ban where the interval [a,b] on which function is defined is divided into n subintervals.

The height of the rectangles is the value of the function calculated at the left-hand end point of the interval containing the base.

The area of a rectangle is base×height.

The approximated area under the curve is sum of the areas of each rectangle.

Calculation:

Consider the provided table,

Time t (hr) N(t)(mg)
0 0
0.5 247.3
1 270
1.5 236.4
2 178.3
2.5 113.9
3 56.2
3.5 19.3

Since, the table represents the corresponding values of y at various x.

Thus, the height of the function from 0 to 3.5 is 0, 247.3, 270, 236.4, 178.3, 113.9, 56.2 respectively.

The base of each rectangle is 0.5.

Recall that the height of the rectangles is the value of the function calculated at the left-hand end point of the interval containing the base.

Record these values in a table.

Rectangle Base left-hand end point Height Area=base×height
[0,0

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