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QuestionDetermine the growth type of an investment account with balances: Year 1: \$1,200, Year 2: \$1,260, Year 3: \$1,323, Year 4: \$1,389.15. Options: A, B, C, D, E.

Studdy Solution

STEP 1

Assumptions1. The account balance at the beginning of each year is given in the table. . No other deposits have been made to the account.
3. We are to determine the type of growth (linear or exponential) and the annual interest rate.

STEP 2

First, we need to calculate the growth rate between the first and second year. We can do this by subtracting the balance of the first year from the balance of the second year, and then dividing by the balance of the first year.
Growth rate=Balanceyear2−Balanceyear1Balanceyear1Growth\, rate = \frac{Balance_{year2} - Balance_{year1}}{Balance_{year1}}

STEP 3

Now, plug in the given values for the balances of the first and second year to calculate the growth rate.
Growth rate=$1,260.00−$1,200.00$1,200.00Growth\, rate = \frac{\$1,260.00 - \$1,200.00}{\$1,200.00}

STEP 4

Calculate the growth rate.
Growth rate=$1,260.00−$1,200.00$1,200.00=0.05Growth\, rate = \frac{\$1,260.00 - \$1,200.00}{\$1,200.00} =0.05

STEP 5

Convert the growth rate to a percentage to get the annual interest rate.
Annual interest rate=Growth rate×100%Annual\, interest\, rate = Growth\, rate \times100\%

STEP 6

Calculate the annual interest rate.
Annual interest rate=0.05×100%=5.00%Annual\, interest\, rate =0.05 \times100\% =5.00\%

STEP 7

Next, we need to determine whether the growth is linear or exponential. If the growth rate is constant from year to year, the growth is linear. If the growth rate is increasing or decreasing from year to year, the growth is exponential.

STEP 8

Calculate the growth rate between the second and third year.
Growth rate=Balanceyear3−Balanceyear2Balanceyear2Growth\, rate = \frac{Balance_{year3} - Balance_{year2}}{Balance_{year2}}

STEP 9

Plug in the given values for the balances of the second and third year.
Growth rate=$,323.00−$,260.00$,260.00Growth\, rate = \frac{\$,323.00 - \$,260.00}{\$,260.00}

STEP 10

Calculate the growth rate.
Growth rate=$,323.00−$,260.00$,260.00=0.05Growth\, rate = \frac{\$,323.00 - \$,260.00}{\$,260.00} =0.05

STEP 11

Since the growth rate is the same as the growth rate between the first and second year, the growth is linear.
Therefore, the account is growing linearly at an annual interest rate of5.00%.
However, to make sure our conclusion is correct, we should also calculate the growth rate between the third and fourth year.

STEP 12

Calculate the growth rate between the third and fourth year.
Growth rate=Balanceyear4−BalanceyearBalanceyearGrowth\, rate = \frac{Balance_{year4} - Balance_{year}}{Balance_{year}}

STEP 13

Plug in the given values for the balances of the third and fourth year.
Growth rate=$,389.15−$,323.00$,323.00Growth\, rate = \frac{\$,389.15 - \$,323.00}{\$,323.00}

STEP 14

Calculate the growth rate.
Growth rate=$,389.−$,323.00$,323.00=0.05Growth\, rate = \frac{\$,389. - \$,323.00}{\$,323.00} =0.05

STEP 15

Since the growth rate is the same as the growth rate between the first and second year, and between the second and third year, the growth is indeed linear.
Therefore, the account is growing linearly at an annual interest rate of5.00%.
The correct answer is B. The account is growing linearly at an annual interest rate of5.00%.

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