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:
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Define the rented area as : The area is in square feet, where must satisfy , since Isabel can rent as little as 100 square feet or as much as 8,100 square feet.
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Calculate the side length of the square: If the area of a square is , the length of one side of the square is .
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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:
Answer:
The total length of a row of bookshelves, , is given by the function: where is the area of the rented space in square feet.
Would you like me to expand on this, such as graphing , discussing its domain/range, or exploring related concepts?
Related Questions:
- What is the domain and range of the function in this context?
- How would the function change if Isabel wanted rectangular spaces instead of square spaces?
- What is the value of if Isabel rents the entire warehouse?
- How would you compute the area if the length of a row of bookshelves was provided instead?
- Can take values like 2500, and what would 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.
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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
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