Part 2: Solve a real-world problem using a square root function.

Question

When Isabel began her book-selling business, she stored her inventory in her garage. Now that her business has grown, she wants to rent warehouse space. Lisa owns a large warehouse nearby and can rent space to Isabel. The area of the warehouse is 8,100 square feet. Lisa is willing to rent Isabel as little as 100 square feet of the space or up to as much as the entire warehouse. Her only requirement is that all spaces must be square.
The total length of each row of bookshelves will be of the length of the storage space.
A. Let x be the area of the space that Isabel rents and f(x) represent the total length of a row of bookshelves. How would you find the length of a row of bookshelves? (3 points)

b. Write a function that expresses f(x).

Answer 1

To solve this problem, we'll follow through with both parts of the question.

Part A

1. Understanding the Area of the Space: If Isabel rents an area $x$ square feet of warehouse space, and since this space must be square, we can express the side length of the square space she rents as $\sqrt{x}$.

2. Relationship Between Area and Length of Bookshelves: According to the problem, the total length of each row of bookshelves will be proportional to the length of the storage space. Therefore, we need to determine how that length relates to the area.

   If each side of the square is $\sqrt{x}$, then the total length of the row of bookshelves would simply be the perimeter or the length of one side since we assume only one row for simplicity (or we can consider it being linearly related).

   Thus, we can consider the function $$f(x)=k\cdot \sqrt{x}$$ where $k$ is a proportionality constant. If the problem states that the total length is simply the length of each side (without any proportionality), then we can directly use: $$f(x)=\sqrt{x}$$

Part B

Using our previous reasoning, we can write the function $f(x)$ that expresses the total length of a row of bookshelves as follows:

$$f(x)=\sqrt{x}$$

This function will take any area $x$ that Isabel rents (from 100 square feet up to 8,100 square feet) and output the corresponding length of a row of bookshelves.

Summary

• Function f(x): $f(x)=\sqrt{x}$
• Where: $x$ is the area (in square feet) of the rented space Isabel chooses for her bookshelves.

This function allows Isabel to determine how long her bookshelves can be based on the area she rents. For example, if she rents 400 square feet, the length of the bookshelf row would be $f(400)=\sqrt{400}=20$ feet.