A mother gives her children two options for their weekly allowance.
Option 1: Increase the allowance by 7% each year.
Option 2: Increase the allowance by $1 each year.
If their allowance is originally $10, which function reveals the difference in allowance between the two options, where x is the number of years?
A.
B.
C.
D.
The increase in allowance for option 1 each year can be calculated using the formula: Increase = Original Allowance * (Percentage Increase/100) = $10 * (7/100) = $0.70
The increase in allowance for option 2 each year is always $1.
Therefore, the difference in allowance between the two options after x number of years can be represented by the function:
Difference = (Increase Option 2 - Increase Option 1) * x = ($1 - $0.70) * x = $0.30 * x
The correct answer is D.
To find the difference in allowance between the two options over x years, we need to calculate the amounts separately for each option and then find the difference.
Option 1: Increase the allowance by 7% each year.
The initial allowance is $10.
After one year, the allowance increases by 7% of $10, which is $0.7.
So, the allowance after one year is $10 + $0.7 = $10.7.
After two years, the allowance increases by 7% of $10.7, which is $0.749.
So, the allowance after two years is $10.7 + $0.749 = $11.449.
In general, the allowance after x years can be calculated using the formula:
Amount = $10 * (1 + 7%)^x
Option 2: Increase the allowance by $1 each year.
The initial allowance is $10.
After one year, the allowance increases by $1.
So, the allowance after one year is $10 + $1 = $11.
After two years, the allowance increases by an additional $1.
So, the allowance after two years is $11 + $1 = $12.
In general, the allowance after x years can be calculated using the formula:
Amount = $10 + $1 * x
Now, let's find the difference between the amounts for each option:
Difference = Amount for Option 1 - Amount for Option 2
Difference = ($10 * (1 + 7%)^x) - ($10 + $1 * x)
Therefore, the correct function that reveals the difference in allowance between the two options, where x is the number of years, is D. ($10 * (1 + 7%)^x) - ($10 + $1 * x)
To find the difference in allowance between the two options, we need to compare the total allowance for each option after x years.
For Option 1, where the allowance increases by 7% each year, the total allowance after x years can be calculated using the formula: 10 * (1 + 0.07)^x.
For Option 2, where the allowance increases by $1 each year, the total allowance after x years can be calculated using the formula: 10 + x.
Now, we need to find the difference in allowance between the two options. The function that represents this difference is simply the result of subtracting the total allowance for Option 1 from the total allowance for Option 2:
10 + x - (10 * (1 + 0.07)^x)
Therefore, the correct answer is D.