Problem 71 - Precalculus: Functions and Graphs

Question

Every cross section of a drainage ditch is an isosceles trapezoid with a small base of 3 feet and a height of 1 foot, as shown in the figure. Determine the width of the larger base that would give the ditch a cross-sectional area of \(5 \mathrm{ft}^{2}\).

Step-by-Step Solution

Answer

The width of the larger base is 7 feet.

Step 1: Understand the Problem

An isosceles trapezoid is described with a smaller base, a larger base, and a height. We need to find the larger base given the area of the trapezoid. The problem specifies a small base of 3 feet, a height of 1 foot, and a total area of 5 ft².

Step 2: Write the Area Formula for a Trapezoid

The area of a trapezoid can be calculated using the formula: \[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]where \( b_1 \) and \( b_2 \) are the lengths of the two bases, and \( h \) is the height.

Step 3: Substitute Known Values

The formula becomes:\[ 5 = \frac{1}{2} \times (3 + B) \times 1 \]where 3 feet is the small base and 1 foot is the height. \( B \) represents the unknown larger base.

Step 4: Simplify the Equation

Multiply both sides by 2 to eliminate the fraction:\[ 10 = (3 + B) \times 1 \]This simplifies to:\[ 10 = 3 + B \].

Step 5: Solve for the Larger Base

Isolate \( B \) by subtracting 3 from both sides:\[ B = 10 - 3 \]\[ B = 7 \].

Step 6: Verify the Solution

Substitute \( B = 7 \) back into the area formula to verify:\[ A = \frac{1}{2} \times (3 + 7) \times 1 = \frac{1}{2} \times 10 = 5 \]The calculation confirms the correct solution.

Key Concepts

Isosceles Trapezoid

An isosceles trapezoid is a special kind of trapezoid where the non-parallel sides, also known as the legs, are equal in length. This symmetry simplifies calculations and gives the trapezoid unique properties:

• The two longer sides (legs) are of equal length.
• It often has line symmetry, meaning it can be split into two mirror-image halves.
• The angles adjacent to each base are equal.

Understanding these characteristics is essential when working with problems involving isosceles trapezoids, like calculating their area or determining unknown lengths. In our problem, knowing that the trapezoid is isosceles helps confirm that the sides and angles are balanced, providing a consistent basis for the calculations.

Cross-Sectional Area

The cross-sectional area is essentially a slice of the 3D shape, in this case, a drainage ditch, serving as a 2D representation that helps us understand the flow capacity or material requirements for that section. The area of the isosceles trapezoid in this problem is given as 5 square feet. This area is crucial because it allows us to link the different dimensions of the trapezoid and constrain one dimension based on the others. Understanding how these dimensions affect the area enables us to solve for unknown values, such as the larger base in this exercise. The formula we used is the trapezoid area formula, which sums the bases, multiplies by the height, and divides by two.

Base of Trapezoid

The base of a trapezoid refers to the parallel sides, in this exercise referred to as the smaller base and the larger base. In a practical sense, the base dimensions determine the width of the drainage ditch across its length. One of the bases is already provided, at 3 feet, and this is the smaller base. To find the larger base, we used the formula for the area of a trapezoid. The sum of both bases accounts for the total width of the shape. Solving for the unknown larger base involved isolating it within the formula, resulting in a calculated value of 7 feet. Adjusting the base sizes while maintaining the same cross-sectional area can affect the ditch's overall appearance and function.

Height of Trapezoid

Height in a trapezoid measures the perpendicular distance between the two parallel bases. It is a fundamental dimension because it dictates the "depth" or vertical stretch of the trapezoid which, in this context, is vital when considering the capacity of the ditch. Here, the height is given as 1 foot. This value was directly used in calculating the area, serving as a multiplier of the average width of the trapezoid (the average of the two bases). Understanding the height helps solve for other properties of the trapezoid, like the area or unknown base length, through the trapezoidal formula, ensuring the ditch is engineered correctly to meet the design specifications.