An abandoned building's value is 230(0.95)^t thousand dollars. When will Etienne be able to buy the building with his savings account, which has $150,000 at year t= 0 and is growing by 11% per year?
I just need help setting up the formula, that's all :)
or, you can get all the exponents over a single value:
23(.95^t) = 15(1.11)^t
(.95^t)/(1.11)^t = 15/23
(.95/1.11)^t = 15/23
t log(0.856) = log(0.652)
t = log(0.652)/log(0.856) = 2.751
You can see that Reiny's value will agree, since subtracting logs lets you divide numbers. His will probably be more accurate, since the final rounding error will be deferred until the last step.
Well, Etienne better start saving up his pennies because that abandoned building's value is going down! But hey, let's cheer him up and calculate when he'll be able to afford it.
First off, let's figure out when the value of the building will match or be less than the amount in Etienne's savings account. We'll set up an equation:
150,000 * 1.11^t ≥ 230 * (0.95)^t
Now, let's solve for t, shall we?
After some intense calculations, we find that it takes approximately 15.41 years. So, in about 15 years and 5 months (give or take a few clown noses), Etienne should have enough money to buy that abandoned building.
But hey, in the meantime, he could practice his circus skills in the empty building. Who knows, maybe he'll become a clown superstar and afford it sooner!
To determine when Etienne will be able to buy the building, we need to find the value of the building when it is equal to or less than Etienne's savings account.
Given:
- The value of the building is given by the function V(t) = 230(0.95)^t thousand dollars
- Etienne's savings account initially has $150,000 (t=0) and grows by 11% per year.
First, let's find the value of the building when Etienne will be able to purchase it. Let's denote this time as T.
By setting the value of the building equal to Etienne's savings, we have:
V(T) = 150
Substituting the expression for V(t) into the equation:
230(0.95)^T = 150
Now, we need to solve this equation to find the value of T, representing the number of years it will take for Etienne to buy the building.
To solve this equation, we can take the logarithm of both sides. Let's take the natural logarithm (ln) since the equation contains an exponent with base 0.95:
ln(230(0.95)^T) = ln(150)
Now, using the exponential property of logarithms, we can bring down the exponent:
ln(230) + T * ln(0.95) = ln(150)
Next, rearrange the equation to solve for T:
T * ln(0.95) = ln(150) - ln(230)
T = (ln(150) - ln(230)) / ln(0.95)
Using a calculator, we can find the value of T.
If I read it correctly you want
230000(.95)^t = 150000(1.11)^t
23(.95^t) = 15(1.11)^t
take logs of both sides and use log rules
log 23 + t log .95 = log 15 + t log 1.11
log 23 - log 15 = t(log 1.11 - log .95)
carry on