Sales at a certain department store follow the model where y is the total sales in thousands of dollars and x is the number of years after 2001. What was the first year that sales fell below $50,000?
what model?
gchm
To find the first year that sales fell below $50,000, we need to solve the equation y < 50, where y represents the total sales in thousands of dollars.
Given the equation y = 100(1.05)^x, where x is the number of years after 2001, we can substitute 50 for y in the equation:
50 = 100(1.05)^x
To solve for x, we need to isolate the variable x on one side of the equation. We can do this by dividing both sides of the equation by 100:
0.5 = (1.05)^x
Now, to solve for x, we need to take the logarithm of both sides of the equation. The natural logarithm (ln) is commonly used in these cases:
ln(0.5) = ln[(1.05)^x]
Using the property of logarithms that states ln(a^b) = b * ln(a), we can simplify the equation:
ln(0.5) = x * ln(1.05)
Next, we need to isolate x by dividing both sides of the equation by ln(1.05):
x = ln(0.5) / ln(1.05)
Using a calculator, we can evaluate this expression:
x ≈ -13.5164
Since x represents the number of years after 2001, and we cannot have a negative number of years, we need to take the absolute value of x:
|x| ≈ 13.5164
Rounding up to the nearest whole number, we get:
|x| ≈ 14
Therefore, the first year that sales fell below $50,000 is 2015.