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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 1.5, Problem 31E

(a)

To determine

**To find:** The amount remaining bismuth-210 after 15 days.

Expert Solution

The amount remaining bismuth-210 after 15 days is 25mg.

**Given:**

The half-life of bismuth-210 is 5 days.

The mass of sample bismuth-210 = 200 mg.

**Calculation:**

Let

That is

After 5 days the bismuth level is

Substitute 200mg for

After 10 days the bismuth level is

Substitute 100mg for

After 15 days the bismuth level is

Substitute 50mg for

Therefore, the amount of bismuth after 15 days is 25mg.

(b)

To determine

**To find:** The amount of bismuth after *t* days.

Expert Solution

After *t* days mass of bismuth is

**Calculation:**

Consider half –life bismuth mass in 5 days =

Where

After
*x* be variable.

Therefore after *t* days mass of bismuth as follows,

Substitute
*x* and 200 for

Therefore, after *t* days mass of bismuth is

(c)

To determine

**To estimate:** The remaining amount of bismuth after 3 weeks.

Expert Solution

After 3 weeks the amount of bismuth is

**Given:**

Consider after 5 days the amount of bismuth = 100 mg.

**Calculation:**

After 5 days the amount of bismuth is

By using part (b) solution
*t* days.

Obtain the amount after 3 weeks that means 21 days.

Substitute 21 for *t* in

Therefore, after 3 weeks the amount of bismuth is

(d)

To determine

**To estimate:** The amount of bismuth reduced to 1mg.

Expert Solution

The amount of bismuth mass reduced to 1mg for 38.2215 days.

**Given:**

The reduced amount of bismuth = 1mg.

**Graph:**

Let *x* axis for days and *y* axis for bismuth (mg).

Using online graphing calculator to draw the graphs for bismuth level and days as shown below Figure 1.

Observe the graph, an initial stage bismuth mass is 200 mg.

Then after 5 days the mass is reducing to 100 mg.

After 10 the mass is reducing to 50 mg.

After 15 days mass is reducing to 25 mg.

Therefore, the mass of bismuth is reducing when the days are increasing.

**Calculation:**

From the graph, the general form of bismuth after *t* days

Substitute 1 for

Take naturel logarithm on both sides,

Simplify further,

Therefore, the amount of bismuth mass reduced to 1mg for 38.2215 days.