Part a.
We have the function:
[tex]y=3000(6^{-0.1x})[/tex]
Notice that, for x = 0, the exponent of 6 is zero, so the result inside the parenthesis is 1. Therefore, for x = 0, y = 3000. This is the y-intercept of the function, i.e., the point where the function touches the y-axis.
This seems to happen for options A and C.
Then, to know which one is correct, let's find y(30):
[tex]y\mleft(30\mright)=3000(6^{-0.1\cdot30})=3000(6^{-3})=\frac{3000}{6^3}=\frac{3000}{216}\cong14[/tex]
Since each step on the y-axis represents 350 units, option C shows that y(30) > 350.
On the other hand, option A shows y(30) above and very close to zero, way below 350, as it should be.
Therefore, option A is correct.
Part b.
Notice that the x-axis shows steps of 5 units. So, to find the weekly sales 10 weeks after the campaign ended, we need to identify the point with x-coordinate equal to 10, and find its y-coordinate:
Notice that the y-coordinate for this point is between 350 and 700.
Also, using x = 0 in the given expression for the function, we find that x = 500:
[tex]y(10)=3000(6^{-0.1\cdot10})=3000(6^{-1})=\frac{3000}{6^{}}=500[/tex]
Therefore, the weekly sales 10 weeks after the campaign ended is $500.
Part c.
We can see from the above results that the weekly sales have declined significantly after the end of the ad campaign: it was $3000 just when the campaign had ended, and have declined to $500 (less than half) 10 weeks after that.
Therefore, the correct answer is option C.
