Question

NO LINKS!! Jacob just won $32000 on the new game show, "The Wall,". He invested his winnings at an interest rate of 4.5%, compounded quarterly. ​

Answer 1

Answers:

a) See the table below

b) The equation is
`y = 32000(1.01125)^x`

c) $40,024.02

d) See the graph below

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Explanations:

a)

Start with part (b) where I detail how to get the equation.

Once the equation is found, plug in x = 0 to get

`y = 32000(1.01125)^x\\\\y = 32000(1.01125)^0\\\\y = 32000\\\\`

Repeat for x = 1

`y = 32000(1.01125)^x\\\\y = 32000(1.01125)^1\\\\y = 32360\\\\`

Repeat for x = 2, x = 3, x = 4 and x = 20 to get the table shown below.

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b)

The template for any exponential equation is
`y = a(b)^x`

a = starting amount = 32000

b = growth factor

The annual interest rate is 4.5%

We compound quarterly, so the quarterly rate is (4.5%)/4 = 1.125% which converts to the decimal form 0.01125; adding one to this leads to the growth factor of b = 1.01125

We go from
`y = a(b)^x` to
`y = 32000(1.01125)^x`

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c)

Plug in x = 20 to represent 20 quarters have elapsed (aka 20/4 = 5 years)

`y = 32000(1.01125)^x\\\\y = 32000(1.01125)^(20)\\\\y \approx 32000(1.25075052084381)\\\\y \approx 40024.016667002\\\\y \approx 40024.02\\\\`

The investment would be worth $40,024.02 after five years, aka twenty quarters.

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d)

See below for the graph. I'm using GeoGebra to make the graph. Another option is Desmos. It's preferable to use technology than to graph by hand. If you wanted to graph by hand, then you'd plot each of the points found in the table. Then draw a curve through all those points.

Answer 2

Compound interest formula

`\sf A=P(1+(r)/(n))^(nt)`

where:

• A = final amount
• P = principal amount
• r = annual interest rate (as a decimal)
• n = number of compounding periods per unit of time
• t = time in decimal years (e.g. 6 months = 0.5 yrs)

Given:

• P = $32,000
• r = 4.5% = 0.045
• n = 4 (compounded quarterly)
• t = 0.25x

`\implies y=32000(1+(0.045)/(4))^(4 * 0.25x)`

`\implies y=32000(1.01125)^x`

Part (a)

`\begin{tabular}c \cline{1-2} \bf Quarterly & \bf \$ \\\cline{1-2} 0 & 32000.00\\\cline{1-2} 1 & 32360.00\\\cline{1-2} 2 & 32724.05\\\cline{1-2} 3 & 33092.20\\\cline{1-2} 4 & 33464.48\\\cline{1-2} --- & --- \\\cline{1-2} 20 & 40024.02\\\cline{1-2}\end{tabular}`

Part (b)

`y=32000(1.01125)^x` (where x is the number of quarterly periods)

Part (c)

$40,024.02

Part (d)

see attached