Problem 36 - Precalculus Mathematics for Calculus

Question

A pasture is twice as long as it is wide. Its area is \(115,200 \mathrm{ft}^{2} .\) How wide is the pasture?

Step-by-Step Solution

Answer

The width of the pasture is 240 feet.

Step 1: Define Variables

Let's denote the width of the pasture as \(w\) (in feet). Since the pasture is twice as long as it is wide, the length is \(2w\).

Step 2: Write the Area Formula

The area \(A\) of a rectangle is calculated using the formula \(A = ext{length} \times ext{width}\). Substituting the expressions for length and width, we get: \[A = 2w \times w = 2w^2\]

Step 3: Set Up the Equation

We know the area of the pasture is \(115,200 \, \mathrm{ft}^2\). So, we set up the equation: \[2w^2 = 115,200\]

Step 4: Solve for \(w^2\)

Divide both sides of the equation by 2 to solve for \(w^2\): \[w^2 = \frac{115,200}{2} = 57,600\]

Step 5: Solve for \(w\)

Take the square root of both sides to solve for \(w\): \[w = \sqrt{57,600}\]. Calculating the square root gives \(w = 240\).

Step 6: Verify the Solution

Substitute \(w = 240\) back into the context: the length is \(2 \times 240 = 480\) feet, and the calculated area is \(480 \times 240 = 115,200 \, \mathrm{ft}^2\), which matches the given area.

Key Concepts

Area of Rectangle

When you're faced with a problem involving the area of a rectangle, it's important to understand the relationship between its length, width, and area. The formula for finding the area is quite straightforward:

• Formula: \(A = ext{length} \times ext{width}\)
• In this formula, \(A\) stands for area, while length and width are the dimensions of the rectangle.

Let's take the pasture in the original problem as an example. You know the area is given as \(115,200 \, \text{ft}^2\). It's stated that the pasture is twice as long as it is wide, making the width \(w\) and the length \(2w\). Plugging these into the area formula gives: \[A = 2w \times w = 2w^2\].This equation incorporates both the length and width, ensuring you have a complete geometric understanding of the rectangle's dimensions.

Equation Solving

Solving equations is a fundamental skill in precalculus and crucial in tackling problems like these, where you're tasked with finding unknown dimensions from given information. Here's how we approach equation solving with the pasture problem:

• We first formulated the equation from our area setup: \(2w^2 = 115,200\).
• To isolate \(w^2\), we divided both sides by 2, resulting in \(w^2 = 57,600\).

Now, you've got a simpler equation to deal with. Solving for \(w\) involves taking the square root of both sides:\[w = \sqrt{57,600}\]Remember, this step means you're looking for the original dimension that, when squared, gives \(57,600\). Calculating the square root finally leads us to: \(w = 240\). Each step in this process methodically transforms the problem, peeling back layers until you isolate the variable you need to find.

Mathematical Reasoning

Mathematical reasoning is all about making logical deductions and verifying that your solutions fit the parameters of the original problem. Here, it's vital to connect the dots:

• First, we identified the relationship between length and width using the problem's condition—that the pasture's length is twice the width.
• Next, we used this relationship to set up our area equation, methodically solving it step-by-step, ensuring logical consistency throughout.

Lastly, upon solving for the width, verifying is crucial. This part of reasoning checks our solution within the problem's given context. Substitute \(w = 240\) back in:

• Length would be \(480\), since \(2 \times 240\).
• Then, the recalculated area is \(480 \times 240 = 115,200\, \text{ft}^2\), which matches perfectly.

Such verification solidifies the accuracy of your reasoning, reinforcing confidence in your solution integrity.