Problem 63 - College Algebra and Calculus: An Applied Approach

Question

A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters (see figure). Within what bounds must the length be?

Step-by-Step Solution

Answer

The length of the rectangular field must be between approximately 15.81 meters and 34.19 meters.

Step 1: Formulate the Constraints

First, write down the formulas for the perimeter and the area of a rectangle. The perimeter \(P\) of a rectangle with length \(L\) and width \(W\) is \(P=2(L+W)\), and the area \(A\) is \(A=LW\). Given that \(P = 100\) m and \(A = 500\) m², we have the two equations \(2(L + W)=100\) and \(LW=500\).

Step 2: Solve for \(W\)

Rearrange the equation for \(P\) to compute for \(W\). From \(2(L+W)=100\), we can simplify to \(L+W=50\) m. Isolate \(W\) to obtain \(W=50-L\).

Step 3: Substitute \(W\) in Area Equation

Substitute \(W=50-L\) into the equation for \(A\), which gives us the equation \(L(50-L)=500\). Expand this to \(50L - L^2 = 500.\)

Step 4: Solve for \(L\)

Rearrange and solve the equation \(50L - L^2 = 500\) giving us a quadratic equation \(L^2 - 50L + 500 = 0\). Solve this quadratic equation for \(L\) using the quadratic formula \(L = \frac{50 \pm \sqrt{50^2 - 4*500}}{2}\), which yields two solutions for \(L\) that will represent the bounds for the length of the rectangle.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra, often represented in the form \( ax^2 + bx + c = 0 \). They arise in many areas such as physics, engineering, and as we see here, geometry. When solving quadratic equations, we commonly encounter terms like \( L^2 - 50L + 500 = 0 \) in this exercise. This form is the "standard form" of a quadratic equation, where:

• \( L^2 \) is the quadratic term,
• \( 50L \) is the linear term,
• 500 is the constant term.

To find solutions, we use the quadratic formula: \[ L = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]. This helps us determine the possible values for the variable \( L \) that satisfy the equation. In our context of rectangular fields, these solutions provide the lengths that meet the constraints for area and perimeter. Quadratic equations yield important real-world solutions, encompassing geometric dimensions and constraints.

Rectangular Geometry

Rectangular geometry encompasses shapes with four sides and right angles, known as rectangles. In mathematics, these figures are characterized by parameters such as length \( L \) and width \( W \). These parameters are essential as they determine properties like perimeter and area.

In the given problem, the playing field's dimensions are subject to certain conditions, like the perimeter \( P = 2(L + W) \) and area \( A = LW \). By manipulating these formulas, we can express one property in terms of the other.

• Perimeter gives us "circumference," or total boundary, of the rectangle.
• We can solve for unknown dimensions when one characteristic, like length, is constrained.

This blending of algebra with geometric formulas reveals how these mathematical principles work together to provide solutions that meet physical demands in real-world spaces.

Perimeter and Area

In any geometry-related problem involving rectangles, understanding the concepts of perimeter and area is crucial.

The perimeter of a rectangle, calculated by \( P = 2(L + W) \), represents the total distance around the shape. For the field in question, the perimeter is 100 meters which limits the sum of the length and width to 50 meters.

• Perimeter is significant in problems where the physical length of fencing, border, or frame material is constrained.
• It influences spatial design and layout considerations.

Area, on the other hand, is the space enclosed within the rectangle's boundaries, found using \( A = LW \). This field has a designated area requirement of at least 500 square meters, meaning the choices for \( L \) and \( W \) must multiply to this number or greater.

• Area affects space utilization and capacity.
• It's pivotal in assessing usage efficiency of rectangular domains.

These calculations allow us to establish bounds for dimensions that meet predefined geometric requirements, harmonizing mathematical theory with practical application.