Question

A parent increases a child’s monthly allowance by 20% each year. If the allowance is $8 per month now, in about how many years will it take to reach $20 per month? Use the equation 20 = 8(1.2)x to solve the problem. Round to the nearest year. 1 year 5 years 2 years 16 years

Answer 1

Answer:

6 years

Step-by-step explanation:

A parent increases a child’s monthly allowance by 20% each year. If the allowance is $8 per month now. This is an exponential function, An exponential function is given by:

$y=ab^x$

Let x be the number of years and y be the allowance. The initial allowance is $8, this means at x = 0, y = 8

$y=ab^x\\8=ab^0\\a=8$

Since it increases by 20% each year, i.e 100% + 20% = 1 + 0.2 = 1.2. This means that b = 1.2

Therefore:

$y=ab^x\\y=8(1.2^x)$

To find the number of years will it take to reach $20 per month, we substitute y = 20 and find x

$20=8(1.2)^x\\20/8=1.2^x\\1.2^x=2.5\\Taking \ natural\ log\ of \ both\ sides:\\ln(1.2^x)=ln2.5\\xln(1.2)=0.9163\\x=0.9163/ln(1.2)\\x=5.026$

x = 6 years to the nearest year