Question

If the value of a cellular phone declines by 33% of its current value every year, approximately how many years for the price of a $243 cell phone to reach $72?

a) 3 years
b) 4 years
c) 5 years
d) 6 years

Answer 1

To determine the number of years for a $243 phone to reach $72, we use the exponential decay formula and solve for $72 = $243 × (0.67)^t, resulting in approximately 3.11 years, which is rounded up to 4 years.

Step-by-step explanation:

To find out approximately how many years it will take for the price of a $243 cell phone to reach $72 with it declining by 33% every year, we can use the formula for exponential decay. The formula is P(t) = P0 × (1 - r)^t, where P(t) is the value of the phone after t years, P0 is the initial value of the phone, and r is the rate of decrease.

• 
  
• Initial value (P0): $243
• 
  
• Rate of decrease (r): 33% or 0.33
• 
  
• Desired value (P(t)): $72

We want to solve for t when P(t) = $72.

Setting up the equation, we have $72 = $243 × (1 - 0.33)^t.

Dividing both sides by $243 gives us 0.2963 = (0.67)^t.

Using logarithms, we can solve for t:
log(0.2963) = t × log(0.67),

t = log(0.2963) / log(0.67) which approximately equals 3.11.

Since we cannot have a fraction of a year in this context, we round up to the next whole number, which gives us 4 years (option b).