Part 1: Emily is a real estate developer who owns a large plot of land with area A acres valued at v dollars per acre. She subdivides the land into n plots of equal area and sells each of them at value for a price P.
1.) Choose the statement that correctly describes how A, n, P, and v vary.
a.) A varies jointly with n and v and inversely with P.
b.) n varies jointly with A and P and inversely with v.
c.) P varies jointly with n and A and inversely with v.
d.) v varies jointly with n and P and inversely with A.
e.) Each of the 4 statements is correct.
f.) None of the statements are correct.
Part 2: Write an equation representing the relationship among A, n, P, and v.
In this problem, it would be helpful to do the Part 2 first, and then use the relationship in Part 2 to answer the question in Part 1.
If Emily divides the plot of land into n plots of equal area, each of these plots will have the area `A/n ` acres.
Since the land is valued at v dollars per acre, each of these plots is valued at `A/n*v` dollars.
Since Emily sells these plots at value, that is, she is not marking it up to get any profit, the price P of each plot is exactly the same as the value of the plot:
`P = A/n*v = (Av)/n`
This is the relationship between P, A, n and v.
From here, we see that P proportional to both A and v, which means that P varies jointly with A and v, and P is inversely proportional to n. This means c) is incorrect.
We can also solve this relationship for other three variables:
`A = (nP)/v`
This means A is varies jointly with n and P and inversely with v, so a) is incorrect as well.
`n` varies jointly with A and v and inversely with P. So, b) is incorrect.
v varies jointly with n and P and inversely with A. This means d) is correct.
The statement d) correctly describes the relationship between the two variables.