please help me!

Question

The value of a home in 2015 was $400,000. Its value has been doubling each decade.

1. If v is the value of the home, in dollars, write an equation for v in terms of d, the number of decades since 2015.

Complete the equation below

v = _______________

Use d as your variable.

Use the ^ symbol to represent an exponent.

For example: 52 should be typed as 5^2

Answer 1

v = v₀e^kt; If t = 0, v = v₀ = 400000, t = 0 corresponds 2015 year. Now if t = 10(2025 year) then v = 800000.

So 2 = 10k, then 2 = e^10k. 10k = ln2, k = ln2/10.

So, v = 400000e^(ln2/10)t_{= 400000·2}^t/10, but t/10 = d

So, v = 400000·2^d

Answer 2

v = 400000(2)^d = 400000(2)^d

Here is why:

v(0) = 400000(2)⁰ = 400000 --> original value

v(1) = 400000(2)¹ = 400000(2) = 800000 --> it doubled

v(2) = 400000(2)² = 400000(4) = 1600000 --> it doubled again

--etc---

Answer 3

Since it is doubling every year, we know that we will be multiplying the original value, $400,000 by 2 for the first decade to equal $800,000. But then the next decade, we will be doubling $800,000 to $1,600,000, which is 4 times the original value, which means we need to use the number of decades as our exponent, so our equation becomes:

400,000 x 2^d

Hope this helps! Let me know if you need any more help!