# The date on which the fish population will be the same as it was on January 1, 2002.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.5, Problem 115E

(a)

To determine

## The date on which the fish population will be the same as it was on January 1, 2002.

Expert Solution

The fish population will be the same as it was on January 1, 2002 after 17yearsonJan1,2019_.

### Explanation of Solution

Given:

The fish population in a certain lake rises and falls according to the formula F=1000(30+17tt2), where F is the number of fish at time t and t is measured in years since January 1, 2020.

Calculation:

Substitute 0 for t in the equation F=1000(30+17tt2), to obtain the date on which the fish population will be the same as it was on January 1, 2002.

F=1000(30+17(0)(0)2)F=1000×30F=30000

Therefore, the initial fish population is 30000.

Substitute 30000 for F in the equation F=1000(30+17tt2).

30000=1000(30+17tt2)30+17tt2=30000100017tt2=3030

Simplify the above equation as follows.

17tt2=0t(17t)=0t=0or17t=0t=0ort=17

So, the population of fish will be same again as it was on January1,2002 after 17yearsonJan1,2019.

Thus, the fish population will be the same as it was on January 1, 2002 after 17yearsonJan1,2019_.

(b)

To determine

### The date by which all the fish in the lake will be dead.

Expert Solution

All the fish in the lake will be dead after 18.612yearsonAug12,2020_.

### Explanation of Solution

Formula used:

The solution of a quadratic equation of the form ax2+bx+c=0,a0 can be obtained by using the quadratic formula x=b±b24ac2a.

Calculation:

The number of fish will be zero when all the fish in the lake are dead.

Therefore, substitute 0 for F in the equation F=1000(30+17tt2).

0=1000(30+17tt2)t217t30=0

Use Quadratic formula to find the value of t.

Substitute 1 for a, 17 for b and 30 for c in the quadratic formula.

t=(17)±(17)24×1×(30)2×1=17±289+1202=17±4092

Simplify the above equation as follows.

t=17±20.222t=17+20.222or t=1720.222t=37.222or 3.222t=18.612or 1.61

The number of years must be positive.

Therefore, all the fish in lake will be dead after 18.612years.

Assuming that there will be 365 days in a year, convert 0.612 years to days.

0.612years×366=224days

That is, all the fish in lake will be dead after 18yearsand 224 days.

By using a calendar, it is obtained that all the fish will be dead on Aug12,2020.

Thus, all the fish in the lake will be dead after 18.612yearsonAug12,2020_.

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