A car's value is declining exponentially. the car is currently 3 years old and has a value of$18000. The car sold for$26000 brand new. how much will the car be worth 5 years from now.
Make an equation:
26000*x^3=18000
x^3=9/13
x is the cube root of 9/13
Plug in the values:
26000*(cube root of 9/13)^5 is around $14086.57
To find out how much the car will be worth 5 years from now, we need to determine the rate at which its value is declining exponentially. The given information tells us that the car is currently 3 years old and has a value of $18,000, and it sold for $26,000 when it was brand new.
To determine the rate of decline, we can use the formula for exponential decay:
V = V0 * e^(-kt)
Where:
V = value at a given time
V0 = initial value (at t=0)
e = base of the natural logarithm (approximately 2.71828)
k = rate of decay
t = time
We can rewrite the formula as:
V = V0 * e^(-kt)
=> V / V0 = e^(-kt)
=> ln(V / V0) = -kt
=> ln(V0 / V) = kt
Now we can use the given information to find the rate of decay. We know that after 3 years, the car's value is $18,000, and when it was brand new, its value was $26,000. Plugging these values into the equation:
ln(26,000 / 18,000) = 3k
Simplifying:
ln(26/18) = 3k
Now we can solve for k:
k = ln(26/18) / 3
Using a calculator, we find that k ≈ 0.2532.
Now that we have the rate of decay (k), we can use the formula to find the value of the car 5 years from now:
V = V0 * e^(-kt)
Plugging in the values:
V = 26,000 * e^(-0.2532 * 5)
Calculating this expression, we find that the car will be worth approximately $8,156.28 five years from now.