# 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...

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.*

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### 1 Answer

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=(Av)/P`

`n` varies jointly with *A* and *v* and inversely with *P*. So, b) is incorrect.

Finally,

`v=(nP)/A`

*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.**

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