## 18 = 3.5(1.12)^n

5.142857 = 1.12^n

take log of both sides and use log rules....

n = log 5.142857/log 1.12 = 14.45

so after 14 years the allowance will be below $18

but after 15 years it will be over $18

Reiny
answered

13 years ago

13 years ago

5.142857 = 1.12^n

take log of both sides and use log rules....

n = log 5.142857/log 1.12 = 14.45

so after 14 years the allowance will be below $18

but after 15 years it will be over $18

Explain Bot
answered

11 months ago

11 months ago

Let's denote the number of years as 'x'.

The parent raises the allowance by 12% each year, which means the child's allowance will increase by 12% of its previous value each year. So, the child's allowance after x years can be calculated using the formula:

Allowance after x years = Initial allowance + (Percentage increase Ã— Initial allowance)

In this case, the initial allowance is $3.50, and the percentage increase is 12%. Thus, the equation becomes:

Allowance after x years = $3.50 + (0.12 Ã— $3.50)

We can simplify this equation as:

Allowance after x years = $3.50(1 + 0.12)

Now, we need to solve for 'x' when the allowance reaches $18. So, we can set up the equation:

$3.50(1 + 0.12)^x = $18

To solve for 'x', we can rearrange the equation as:

(1 + 0.12)^x = $18 / $3.50

Now, we can solve this equation using logarithms, specifically the natural logarithm (ln). Taking the natural logarithm of both sides, we have:

ln[(1 + 0.12)^x] = ln($18 / $3.50)

Now, we can simplify further:

x ln(1 + 0.12) = ln($18 / $3.50)

Finally, to isolate 'x', we divide both sides of the equation by ln(1 + 0.12):

x = ln($18 / $3.50) / ln(1 + 0.12)

Using a calculator, we can evaluate the right side of the equation to find 'x'.