Question

Marilee takes all the money from her piggy bank and puts it into a savings account at her local bank. The bank promises an annual interest rate of 2.5% on the balance, compounded semiannually. How much will she have after one year if her initial deposit was $400?

A) $390.06
B) $420.25
C) $410.06
D) $420.50

Which situation would be represented by a discrete probability distribution?
A) the thickness of an item
B) the lifespan of a fruit fly
C) the temperature of a solution
D) number of complaints a company receives in a day

The amount of money in a bank account that is compounded yearly can be represented by the function A(y) = P(1 + r)y, where P is the amount initially deposited, r is the annual interest rate expressed as a decimal, and y is the number of years that have passed since the initial deposit. $2,700 was deposited 14 years ago into a bank account that is compounded yearly, and no additional deposits or withdrawals have been made. If the amount of money now in the bank account is $7,930.42, what is the annual interest rate?
A) about 5%
B) about 6%
C) about 7%
D) about 8%

Which type of data is BEST described by a continuous model?
A) points scored
B) pages in a book
C) children in a family
D) temperature of an engine

Answer 1

C) $410.06; D) number of complaints a company receives in a day ; D) about 8% ; D) temperature of an engine

Explanation:

The equation for compound interest is

`y=p(1+(r)/(n))^(nt)`, where p is the amount of principal, r is the interest rate, n is the number of times the interest is compounded yearly, and t is the number of years.

Our principal is 400; r is 2.5%, which is 0.025 (2.5% = 2.5/100 = 0.025); n is 2 since it is compounded semiannually; and t is 1:

`y=400(1+(0.025)/(2))^(2* 1)\\\\=400(1+0.0125)^2\\\\=400(1.0125)^2=410.0625 \approx 410.06`

A discrete probability distribution will be of something that has distinct values, with no values inbetween. The thickness of an item has fractions and decimals, not distinct values. The lifespan of a fruitfly can be in hours, which can be broken down to minutes and seconds; these are not distinct values. The temperature of a solution can be expressed in decimal form; these are not distinct values. However, the number of complaints a company receives in a day has to be distinct values. You can have 1 complaint or 2, but not 1.5. This means the answer is D.

Plugging our numbers into our equation, we have

`7930.42=2700(1+r)^(14)`

Dividing both sides by 2700,

`(7930.42)/(2700)=(2700(1+r)^(14))/(2700)\\\\2.937193=(1+r)^(14)`

Next we take the 14th root of each side, to cancel the exponent of 14. To do this we can raise it to the 1/14 power:

`(2.937193)^{(1)/(14)}=((1+r)^(14)})^{(1)/(14)}\\\\1.079999984=1+r`

Subtract 1 from each side:

1.079999984-1 = 1+r-r

0.079999984 = r

0.08 ≈ r

This is about 8%.

A continuous model includes not only distinct values but values between them. The number of points scored would only be distinct values, no values between; you will not score 2.5 points, only 2 or 3. The pages in a book are distinct; you cannot have 2.5 pages, either 2 or 3. The number of children in a family is distinct; you cannot have 2.5 children, either 2 or 3. However the temperature of an engine can be different fractions or decimals between distinct temperatures; this makes it continuous.

Answer 2

A=400×(1+0.025÷2)^(2)
A=410.06

r=((7,930.42÷2,700)^(1÷14)−1)×100
r=8%