A business owner opens one store in town A. The equation p(x)=10,000(1.075)^t represents the anticipated profit after t years. The business owner opens a store in town B six months later and predicts the profit from that store to increase at the same rate. Assume that the initial profit from the store in town B is the same as the initial profit from the store in town A. At any time after both stores have opened, how does the profit from the store in town B compare with the profit from the store in town A?
65%
96%
104%
or 154%
It’s 96%
so what is it lol
To compare the profit from the store in Town B with the profit from the store in Town A, we need to find the profit equations for both stores and evaluate them at the same time.
The given equation for the profit from the store in Town A is p(x) = 10,000(1.075)^t.
Since the store in Town B opens six months later than the store in Town A, we need to adjust the time variable t accordingly. The equation for the profit from the store in Town B would be p(x) = 10,000(1.075)^(t - 0.5). Here, we subtract 0.5 from the time, which represents six months or half a year.
Now, let's compare the profit from both stores after they have been open for the same amount of time. Let's assume we want to evaluate the profit after t years.
For the store in Town A, the profit equation is p(x) = 10,000(1.075)^t.
For the store in Town B, the profit equation is p(x) = 10,000(1.075)^(t - 0.5).
Now, evaluate both equations at t years:
Profit from Town A = 10,000(1.075)^t
Profit from Town B = 10,000(1.075)^(t - 0.5)
Comparing the two equations, let's find the ratio of the profit from Town B to the profit from Town A:
Profit from Town B / Profit from Town A = (10,000(1.075)^(t - 0.5)) / (10,000(1.075)^t)
Now, simplify the equation:
Profit from Town B / Profit from Town A = (1.075)^(t - 0.5) / (1.075)^t
Since both stores initially have the same profit, the 10,000 amount cancels out:
Profit from Town B / Profit from Town A = (1.075)^(t - 0.5) / (1.075)^t = (1.075)^(t - 0.5 - t) = (1.075)^(-0.5)
Simplifying further:
Profit from Town B / Profit from Town A = 1 / (1.075^0.5)
Now, calculate the value of 1.075^0.5 using a calculator or by taking the square root of 1.075:
Profit from Town B / Profit from Town A = 1 / 1.0356
Finally, evaluate the ratio:
Profit from Town B / Profit from Town A ≈ 0.9658
Therefore, the profit from the store in Town B is approximately 96.58% of the profit from the store in Town A.
So, the correct answer is approximately 96%.