Direct vs. Inverse Variation
Also this one:
You purchase a new SUV; as age increases, the resale price decreases.
I believe this would be inverse, but the answer packet says neither direct nor inverse.
As I understand inverse, as one value goes up, the other goes down, which is exactly what this is doing???
you are right that value goes down with age
but
not directly or inversely
value is not equal to k t
and
value is not equal to k / t
in fact I need calculus to do it
change in V = k V * change in time
where k is less than one but positive, the rate of change is fast when new
or
dV/V = k dt
ln V = k t +constant
e^lnV = V = constant e^kt
constant is value when t = 0
V = Vi e^kt
if k is -.5
V = Vi e^-.5t for example
This is called exponential decay, in this case of car value but it could be a radioactive element losing a fraction of its protons every year.
You are correct in your understanding of inverse variation, where one value increases as the other decreases. In the case of the resale price of your SUV decreasing as its age increases, this does indeed exhibit an inverse variation. Therefore, it seems that the information in the answer packet is incorrect.
Direct variation and inverse variation are two types of relationships between two variables. In direct variation, when one variable increases, the other variable also increases proportionally. In inverse variation, when one variable increases, the other variable decreases proportionally.
In the case of the resale price of an SUV decreasing as its age increases, it initially seems like an inverse variation, as you mentioned. However, it is important to consider the specific definition of inverse variation.
Inverse variation is characterized by a relationship where the product of the two variables remains constant. In other words, if you multiply the value of one variable by the value of the other variable, the result stays the same.
In the case of the SUV's resale price and age, it is unlikely that the product of the resale price and age would remain constant. As age increases, the resale price generally decreases, but the rate at which it decreases is not necessarily constant. Therefore, it does not exhibit a true inverse variation.
Instead, this relationship could be better described as a general negative correlation. As the SUV's age increases, the resale price tends to decrease, but it does not strictly follow a direct or inverse variation pattern. The rate at which the resale price decreases may vary depending on various factors such as the condition, market demand, or other external factors.
It is worth noting that these types of relationships can sometimes be oversimplified when discussing mathematical concepts. In real-world scenarios, relationships between variables can be more complex and may not perfectly fit into the predefined categories of direct or inverse variation.