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Math Help - Intro Calc HW

  1. #1
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    Intro Calc HW

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    if $10000 is invested at an interest rate compounded annually the investment will grow to $P after x years where

    P=10 000(1.08)^x

    d) how long will it take for the ionvestment to grow $50 000 if the interest is 14% for the first 8 yrs and 10% thereafter?

    e) find the annual interest rate necessary for the $10 000 to double in value in 5 yrs. (give ur answer as percentage)

    Thankyu
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  2. #2
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    Hello, Fibonacci!

    If $10000 is invested at r% compounded annually.
    the investment will grow to $P after x years where: . P\:=\:10 000(1+r)^x

    d) How long will it take for the investment to grow $50 000
    if the interest is 14% for the first 8 yrs and 10% thereafter?

    The $10,000 is invested at 14% for the first 8 years.
    . . Its value will be: . 10,000(1.14)^8 dollars.

    Then that amount is invested at 10% for x years.
    . . The final value is: . 10,000(1.14)^8 \cdot(1.10)^x
    . . which is to equal $50,000.

    There is our equation . . . 10,000(1.14)^8(1.1)^x \:=\:50,000

    Divide by 10,000: . (1.14)^8(1.1)^x \:=\:5

    Divide by (1.14)^8\!:\quad(1.1)^x \:=\:\frac{5}{1.14^8}

    Take logs: . \ln(1.1)^x \:=\:\ln\left(\frac{5}{1.14^8}\right) \quad\Rightarrow\quad x\!\cdot\!\ln(1.1) \:=\:\ln\left(\frac{5}{1.14^8}\right)

    Therefore: . x \;=\;\frac{\ln\left(\frac{5}{1.14^8}\right)}{\ln(1  .1)} \;=\;5.88826729 \;\approx\;6 years.

    It will take: . 8 + 6 \:=\:\boxed{14\text{ years}}




    e) Find the annual interest rate necessary for the $10 000
    to double in value in 5 yrs. (Give your answer as percentage.)
    Let r = the required interest rate.

    Then: . 10,000(1 + r)^5 \;=\;20,000

    Divide by 10,000: . (1+r)^5 \:=\:2

    Take the fifth root: . 1 + r \:=\:\sqrt[5]{2}

    Therefore: . r \;=\;\sqrt[5]{2}-1 \;=\;0.148698355 \;\approx\;\boxed{14.9\%}

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