In 1974, Johnny Miller won 8 tournaments on the PGA tour and accumulated $353,022 in official season earnings. In 1999, Tiger Woods accumulated $6,616,585 with a similar record.
(a) Suppose the monthly inflation rate from 1974 to 1999 was 0.0079. Use the compound interest formula to estimate the equivalent value of Miller's winnings in the year 1999. (Round your answer to the nearest cent.) I got the correct answer of 3741410.25
(b) Find the annual interest rate needed for Miller's winnings to be equivalent in value to Woods's winnings. (Round your answer to two decimal places.)---this is the part I'm having problems with.
The perimeter of a rectangle is 50 inches, and the area is 126 square inches. Find the length and width of the rectangle.
To determine the annual interest rate needed for Miller's winnings to be equivalent to Woods's winnings, we can use the compound interest formula in reverse.
Let's denote the original amount as P (Miller's winnings in 1974), the number of years as t (1999 - 1974), the monthly inflation rate as r, and the final amount as A (Woods's winnings in 1999).
(a) We have already calculated the equivalent value of Miller's winnings in 1999 as $3,741,410.25. So, we have P = $3,741,410.25.
The number of months from 1974 to 1999 is 25 years * 12 months/year = 300 months.
The monthly inflation rate is given as r = 0.0079.
Using the compound interest formula:
A = P * (1 + r)^t
Plugging in the values:
$6,616,585 = $3,741,410.25 * (1 + 0.0079)^300
Now, we can solve for the annual interest rate r.
Dividing both sides of the equation by $3,741,410.25:
$6,616,585 / $3,741,410.25 = (1 + 0.0079)^300
Taking the 300th root of both sides of the equation:
(1.7680)^(1/300) = 1 + r
Subtracting 1 from both sides:
(1.7680)^(1/300) - 1 = r
Converting to a percentage:
r ≈ (1.0496 - 1) * 100 ≈ 4.96%
Therefore, Miller's winnings would need to grow at an annual interest rate of approximately 4.96% for it to be equivalent to Woods's winnings.
(b) The annual interest rate needed for Miller's winnings to be equivalent in value to Woods's winnings is approximately 4.96%.
To find the annual interest rate needed for Miller's winnings to be equivalent in value to Woods's winnings, we can use the compound interest formula in reverse. The formula is:
Equivalent Value = Principal * (1 + Interest Rate)^Time
Where:
- Equivalent Value is the value we want to find (in this case, Woods's winnings).
- Principal is the initial value (in this case, Miller's winnings).
- Interest Rate is the annual interest rate.
- Time is the number of years between the two winnings.
We know the values for Principal (Miller's winnings), Equivalent Value (Woods's winnings), and Time (25 years). Let's solve for the Interest Rate:
Rewriting the formula with the known values:
6600000 = 353022 * (1 + Interest Rate)^25
Divide both sides of the equation by 353022:
6600000/353022 = (1 + Interest Rate)^25
Taking the 25th root on both sides:
(6600000/353022)^(1/25) = 1 + Interest Rate
Subtracting 1 from both sides:
(6600000/353022)^(1/25) - 1 = Interest Rate
Calculating the value on the left side gives:
Interest Rate ≈ 0.0879
So the annual interest rate needed for Miller's winnings to be equivalent to Woods's winnings is approximately 8.79%.