Chapter 9.9, Problem 7CYU

a.

To determine

To find a function to represent value of nickel after t years.

Answer to Problem 7CYU

  $f(t)=20{\left(1+\frac{4.5}{100}\right)}^t$

Explanation of Solution

Given information:

Collector pays $20 for nickel coin

Value of nickel coin increase by 4.5% per year

Formula used:

Price after t years

  $=P{\left(1+\frac{r}{100}\right)}^t$

P is initial price

r rate of increase of price

t is time in years

Calculation:

Initial price $=20$

Rate of increase of price in t years $=4.5$

Using formula to find function of price after t years

  $f(t)=20{\left(1+\frac{4.5}{100}\right)}^t$

b.

To determine

To find a function to represent value of quarter coin after t years.

Answer to Problem 7CYU

  $g(t)=42+2t$

Explanation of Solution

Given information:

Collector pays $42 for quarter coin

Value of quarter increase by $2 per year

Calculation:

Initial price $=42$

Rate of increase of price in t years $=\$2$

Using formula to find function of price after t years

  $g(t)=42+2t$

c.

To determine

To find a function to represents $V(t)=f(t)+g(t)$ .

Answer to Problem 7CYU

  $V(t)=20{\left(1+\frac{4.5}{100}\right)}^t+42+2t$

Explanation of Solution

Given information:

Collector pays $20 for nickel

Value of nickel increase by 4.5% per year Collector pays $42 for quarter

Value of quarter increase by $2 per year

Calculation:

function of price of nickel after t years

  $f(t)=20{\left(1+\frac{4.5}{100}\right)}^t$

Initial price $=42$

Rate of increase of price in t years $=\$2$

Function of price of quarter after t years

  $g(t)=42+2t$

  $V(t)=f(t)+g(t)$

Substituting value of price of nickel and quarter in above

  $V(t)=20{\left(1+\frac{4.5}{100}\right)}^t+42+2t$

This represents total value of nickel and quarter after t years

d.

To determine

To find the value of coins .

Answer to Problem 7CYU

Value of nickel coin after 5 years

  $f(5)=25$

Value of quarter coin after 5 years

  $g(5)=52$

Total value of coins after 5 years

  $V(5)=77$

Explanation of Solution

Given information:

Value of nickel coin after t years

  $f(t)=20{\left(1+\frac{4.5}{100}\right)}^t$

Value of quarter coin after t years

  $g(t)=42+2t$

Total value of coins after t years

  $V(t)=20{\left(1+\frac{4.5}{100}\right)}^t+42+2t$

Calculation:

Value of nickel coin after 5 years

  $f(5)=20{\left(1+\frac{4.5}{100}\right)}^5$

On solving,

  $f(5)=20{\left(1.045\right)}^5$

On solving,

  $f(5)=20{\left(1.045\right)}^5$

On solving,

  $f(5)\approx 24.9$

Approximate

  $f(5)=25$

Value of quarter coin after 5 years

  $g(5)=42+2(5)$

On solving,

  $g(5)=52$

Total value of coins after 5 years

  $V(5)=f(5)+g(5)$

On solving,

  $V(5)=25+52$

On solving,

  $V(5)=77$