Math Problem Statement

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?

Solution

To find the total length of a row of bookshelves, we need to consider the geometric properties of the storage space Isabel rents. Since Lisa requires all rented spaces to be square-shaped, the length of one side of the square can be calculated as the square root of the area.

Step-by-step process:

  1. Define the rented area as xx: The area xx is in square feet, where xx must satisfy 100x8100100 \leq x \leq 8100, since Isabel can rent as little as 100 square feet or as much as 8,100 square feet.

  2. Calculate the side length of the square: If the area of a square is xx, the length of one side of the square is x\sqrt{x}.

  3. Interpret the total length of a row of bookshelves: Since the total length of each row of bookshelves corresponds to the side length of the rented space, we have: f(x)=x.f(x) = \sqrt{x}.

Answer:

The total length of a row of bookshelves, f(x)f(x), is given by the function: f(x)=x,f(x) = \sqrt{x}, where xx is the area of the rented space in square feet.


Would you like me to expand on this, such as graphing f(x)f(x), discussing its domain/range, or exploring related concepts?

Related Questions:

  1. What is the domain and range of the function f(x)=xf(x) = \sqrt{x} in this context?
  2. How would the function change if Isabel wanted rectangular spaces instead of square spaces?
  3. What is the value of f(x)f(x) if Isabel rents the entire warehouse?
  4. How would you compute the area if the length of a row of bookshelves was provided instead?
  5. Can xx take values like 2500, and what would f(x)f(x) be in that case?

Tip:

When working with square shapes, remember that the area and side length are related through the square root function, which is always non-negative.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Geometry
Square Properties
Square Roots

Formulas

f(x) = √x

Theorems

Square root property of area and side length

Suitable Grade Level

Grades 6-8