Problem 35 - Precalculus Mathematics for Calculus

Question

A rectangular garden is 25 ft wide. If its area is \(1125 \mathrm{ft}^{2},\) what is the length of the garden?

Step-by-Step Solution

Answer

The length of the garden is 45 ft.

Step 1: Understand the Relationship Between Length, Width, and Area

The area of a rectangle is calculated by multiplying its length by its width. The formula is given as:\[\text{Area} = \text{Length} \times \text{Width}\] Given that the width of the garden is 25 ft and the area is 1125 ft², we need to find the length.

Step 2: Rearrange the Area Formula to Solve for Length

To find the length, we rearrange the area formula to solve for length:\[\text{Length} = \frac{\text{Area}}{\text{Width}}\] This arrangement isolates the length on one side of the equation.

Step 3: Substitute the Given Values into the Formula

Substitute the given values for area and width into the rearranged formula:\[\text{Length} = \frac{1125 \text{ ft}^2}{25 \text{ ft}}\]This replaces the placeholders in the formula with the actual given values.

Step 4: Calculate the Length

Now perform the division to find the length:\[\text{Length} = \frac{1125 \text{ ft}^2}{25 \text{ ft}} = 45 \text{ ft}\]Thus, the length of the garden is 45 ft.

Key Concepts

Concept 1: Area of a Rectangle

The area of a rectangle is a fundamental concept in geometry that helps us understand the space within a rectangular shape. It is found by multiplying the length of the rectangle by its width. Think of it as counting the number of square units that fit perfectly inside the rectangle. If a rectangle is 4 units long and 3 units wide, then the area is 4 times 3, which equals 12 square units. It's like arranging 12 little squares to fill the rectangle completely. To use this concept for our garden, we were given the area, which is 1125 square feet, and the width, which is 25 feet. Using the area formula helps us work backward to find any missing dimension, like the length, when the other dimensions are known.

Concept 2: Length and Width Relationship

In the context of a rectangle, the length and width are directly related to the area. They are the two dimensions that, when multiplied together, determine how much space the rectangle actually covers. In our specific garden example, the width was given as 25 feet. This tells us only one part of the rectangular shape but locks in a crucial variable. Knowing either the width or the length allows us to use the relationship formula to solve for the missing measurement, provided the area is known. Understanding this relationship is key. It guides us in knowing that you can always find one if the others are given, as the rectangle's proportions are consistent when resembling any rectangular area calculation.

Concept 3: Formula Manipulation

Formula manipulation involves changing the arrangement of equations to solve for unknown variables. In the rectangular garden example, we needed to find the length instead of the area, so we rearranged our formula:The original formula is:\[\text{Area} = \text{Length} \times \text{Width}\]To find the length, we want length by itself, leading us to reorganize the formula like this:\[\text{Length} = \frac{\text{Area}}{\text{Width}} \]This process of rearranging lets us strategically solve problems by using algebra. It's about logically isolating what you don't know using what you do know. In our problem, substituting the known values (1125 ft² for the area and 25 ft for the width) into the rearranged formula gives us the missing length as 45 feet.