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aksik [14]
3 years ago
13

The Johnson family wants to start a college fund for their daughter Gabriella. They put $63, 000 into an account that grows at a

rate of 2.55% per year, compounded quarterly. Given the function G(t)= 63,000(1+ .0255/4)^ 4t, where G(t) represents the amount of money in the account t years after the account is opened, given that no more money is deposited into or withdrawn from the account.
Calculate the number of years it will take for the account to reach approximately $150,000, to the nearest hundredth of a year.
Mathematics
1 answer:
Rainbow [258]3 years ago
5 0

Answer:

<em>It will take approximately 34.13 years</em>

Step-by-step explanation:

The function G(t) below represents the amount of money in some account t years after the account is opened for The Johnson's daughter Gabriella:

G(t)= 63,000(1+ .0255/4)^ {4t}

It's required to find the number of years (t) it will take for the account to reach G(t)=150,000. We need to solve the equation:

63,000(1+ .0255/4)^ {4t}=150,000

Dividing by 63,000 and simplifying:

\displaystyle (1+ .0255/4)^ {4t}=\frac{150,000}{63,000}=2.38095

Taking logarithms:

\displaystyle \log(1+ .0255/4)^ {4t}=\log 2.38095

Applying logarithms property:

\displaystyle (4t) \log(1+ .0255/4)=\log 2.38095

Solving for t:

\displaystyle 4t =\frac{\log 2.38095}{\log(1+ .0255/4)}

\displaystyle t =\frac{\log 2.38095}{4\log(1+ .0255/4)}

Calculating:

\displaystyle t =\frac{0.37675}{0.01104}

\boxed{t \approx 34.13}

It will take approximately 34.13 years

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The price of T-shirts before the sale = 42 euros

The price of scarves before the sale = -77/2 euros

Step-by-step explanation:

The cost of the items Zina buys from the store initially are;

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The reduction in the price of scarves a week later during sales = 1.5 euros

The reduction in the price of T-shirts a week later during sales = 2 euros

The items Zina bought for 40 euros at the reduced price of the sales opportunity are;

Four scarves and five T-shirts for 40 euros

Let 'x' represent the initial price of a scarf and let 'y' represent the initial price of a T-shirt, we  have;

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Expanding equation (2) gives;

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4·x + 5·y = 40 + 6 + 10 = 56

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By multiplying equation (1) by 2 and subtracting the result from equation (3), we get;

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The price of T-shirts before the sale = 12 euros

The price of scarves before the sale = 6.5 euros.

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