The monthly rent for office space in a certain office building is directly proportional to the size of the space. A 687 square foot office rents for $8,244 per month. Let x = number of square feet, y = rent (in dollars per month), and k = the constant of variation. c) Find the equation of variation.


Given that the rent is directly proportional to the size of the office space, the relationship between the rent $ y $ and the number of square feet $ x $ can be expressed as:

\[
y = kx
\]

Where:
- $ y $ is the rent (in dollars per month),
- $ x $ is the size of the office (in square feet), and
- $ k $ is the constant of variation.

From the problem, we know:
- $ y = 8,244 $ dollars for 687 square feet ($ x = 687 $).

### Step 1: Solve for the constant of variation $ k $
Using the formula $ y = kx $, we substitute the values for $ y $ and $ x $:

\[
8,244 = k \cdot 687
\]

Now, solve for $ k $:

\[
k = \frac{8,244}{687} \approx 12
\]

So, the constant of variation $ k $ is 12.

### Step 2: Write the equation of variation
Now that we have $ k = 12 $, the equation of variation is:

\[
y = 12x
\]

This is the equation that relates the rent $ y $ to the size of the office space $ x $.

Would you like further details or have any questions about this?

### 5 Related Questions:
1. What would be the rent for an office space of 800 square feet using the equation $ y = 12x $?
2. How would the rent change if the size of the office space doubled?
3. If the rent for a particular office is $10,000, how many square feet is the office space?
4. How can we interpret the constant of variation $ k $ in practical terms?
5. What would happen to the equation if the relationship was not direct but inverse?

### Tip:
Always check the units when solving real-world variation problems to ensure you're applying the correct proportional relationship.

The monthly rent for office space in a certain office building is directly proportional to the size of the space. A 687 square foot office rents for $8,244 per month. Let x = number of square feet, y = rent (in dollars per month), and k = the constant of variation. Find the equation of variation.

Discover how to find the equation of direct variation for office rent based on the size of the space. Given that a 687 square foot office rents for $8,244, this example explains the calculation of the constant of variation and the final equation y = 12x, suitable for Grades 7-10.

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