Growth Rates: Joe Maur, a catcher for the Minnesota Twins is expected to hit 15 home runs in 2018. If his home-run-hitting ability is expected to grow by 12 percent every year for the following five years, how many home runs is he expected to hit in 2023?
The correct answer and explanation is :
To determine how many home runs Joe Mauer is expected to hit in 2023, given a 12% growth rate each year from 2018, we can use the compound growth formula. The formula for compound growth is:
[
A = P \times (1 + r)^t
]
Where:
- (A) is the amount after time (t),
- (P) is the initial value (in this case, 15 home runs in 2018),
- (r) is the growth rate per year (12%, or 0.12),
- (t) is the number of years of growth (from 2018 to 2023, which is 5 years).
Step 1: Plugging in the known values
We know:
- (P = 15),
- (r = 0.12),
- (t = 2023 – 2018 = 5).
Now, we can substitute these values into the compound growth formula:
[
A = 15 \times (1 + 0.12)^5
]
[
A = 15 \times (1.12)^5
]
Step 2: Calculate the growth factor
Next, we calculate ( (1.12)^5 ):
[
(1.12)^5 \approx 1.7623
]
Step 3: Calculate the final value
Now, we multiply the initial number of home runs by the growth factor:
[
A = 15 \times 1.7623 \approx 26.43
]
Step 4: Round the result
Since home runs are whole numbers, we round 26.43 to the nearest whole number, which is 26.
Conclusion:
Therefore, Joe Mauer is expected to hit 26 home runs in 2023, assuming his home-run-hitting ability grows at a constant rate of 12% per year.
Explanation:
This calculation is based on the concept of compound growth, where each year’s increase is applied to the previous year’s total. Starting with 15 home runs in 2018, Mauer’s home-run output grows by 12% each year. Over the course of five years, this compounding effect leads to an expected home-run total of 26 in 2023.