writing an exponential equation
The function gives the value of a home in dollars in the year 1980 + t. Write an equation that can be used to determine the year in which the value of the home reached $100,000? Use ^ to denote an exponent. Do not solve the equation and do not leave any spaces in your answer.
The equation cannot be solved unless information on the initial value of the home and the rate of appreciation (increase in value) is provided.
Let Vo be the value of the home in 1980.
Let V(t) be the value of the home in year 1980+t. Let a be the rate of appreciation.
V = Vo (1 + a)^t
When V = 100,000
100,000 = Vo (1+a)^t
That equation can be solved for t by using logarithms, but you asked me not to.
What would the answer be?
A typial exponential formula looks like f(t)=base*e^(rt) where r is some rate and t is a time. The base would be the value when t=0.
Suppose the base is 50,000 and r=.07, then
f(t)=50,000*e^(.07*t)
To find t when f(t) = 100,000 we would have
f(t)=100,000= 50,000*e^(.07*t) or
2 =e^(.07*t) or ln2=.07*t or
ln2/.07 = t=9.9 or approx 10 years. Thus at that rate it would take nearly 10yrs to double.
I solved the equation, but your instructions asked you not too.
Thus you should have
2*base=base*e^(rt) or 2=e^(rt) or
ln2/r = t
I worked the example to show how you'd do this.
To write an exponential equation to determine the year in which the value of the home reached $100,000, we need to know the initial value of the home and the rate of appreciation.
Let's call the initial value of the home Vo and the rate of appreciation a. The equation can be written as:
V(t) = Vo * (1 + a)^t
Where V(t) represents the value of the home in the year 1980 + t.
If we want to find the year when the value reached $100,000, we can set V(t) equal to $100,000:
100,000 = Vo * (1 + a)^t
However, without knowing the initial value of the home and the rate of appreciation, we cannot solve the equation. We would need more information to accurately determine the value of t, which represents the number of years it takes for the home's value to reach $100,000.
If you have the initial value of the home (Vo) and the rate of appreciation (a), you can substitute those values into the equation and solve for t using logarithms. But since you specifically asked not to solve the equation, I cannot provide an exact answer.