Question

Question 9(Multiple Choice Worth 5 points) (07.03 MC) The depreciating value of a semi-truck can be modeled by y = Ao(0.83)x, where y is the remaining value of the semi, x is the time in years, and it depreciates at 17% per year. An exponential function comes down from the positive infinity and passes through the points zero comma seventy thousand. The graph is approaching the x axis. What is the value of the truck initially, Ao, and how would the graph change if the initial value was only $50,000? $70,000, and the graph would have a y-intercept at 50,000 $60,000, and the graph would have a y-intercept at 70,000 $70,000, and the graph would fall at a slower rate to the right $70,000, and the graph would fall at a faster rate to the right

Answer 1

The correct answer is $70,000 and the graph would have a y-intercept of $50,000.

The point (0, 70000) is on the y-axis; it is where the data crosses the y-axis, the y-intercept; it is the amount of the vehicle before any depreciation has taken place. This means the initial value was $70,000.

If the truck had an initial value of $50,000, then (0, 50000) would be the y-intercept. The rate of depreciation would not change, so the graph would not change other than this vertical shift.

Answer 2

Two choices are right:

• $70,000, and the graph would have a y-intercept at 50,000, and
• $70,000, and the graph would fall at a slower rate to the right

Step-by-step explanation:

1) Model:

`y=A_0(0.83)^(x)`

2) Given point in the graph: (0, 70000)

3) Initial value means x = 0. Hence, the ordered pair (0, 70000) directly tells you that the initial value is 70,000.

You may also subsititute the value x = 0 in the formula, to get A₀, which is the initial value.

`y=A_0(0.83)^0=70,000\\ \\ A_0(1)=70,000\\ \\ A_0=70,000`

4) If the initial value were only $50,000 instead of $ 70,000, the graph would have a y-intercept at 50,000, because, as explained, the initial value is the value of the function when x = 0, which is the y-intercept.

With that, you get the answer: the initial value is $70,000, and, if the initial value were only $50,000, the graph would have a y-intercept at 50,000

5) As for the rate at which the graph would fall, this is the slope of the function.

If you draw the graphs, you can also see that, although to the left they seem parallel, it is evident that they get closer to each other to the right. Since, the graph when the initial value is $70,000 starts upper, and they get closer to the right, you conclude that if the initial value were lower the graph would fall at a slower rate to the right.

Therefore, this makes the third choice, $70,000, and the graph would fall at a slower rate to the right, also true.