Problem 2 - Technical Mathematics with Calculus

Question

Find the area of (a) a parallelogram whose base is \(4.52 \mathrm{ft}\) and whose altitude is \(2.95 \mathrm{ft}\). (b) a parallelogram whose base is \(16.3 \mathrm{m}\) and whose altitude is \(22.6 \mathrm{m}\) (c) a rhombus whose base is \(14.2 \mathrm{cm}\) and whose altitude is \(11.6 \mathrm{cm}\) (d) a rhombus whose base is 382 in. and whose altitude is 268 in. (e) a trapezoid whose bases are \(3.83 \mathrm{m}\) and \(2.44 \mathrm{m}\) and whose altitude is \(1.86 \mathrm{m}\) (f) a trapezoid whose bases are \(33.6 \mathrm{ft}\) and \(24.7 \mathrm{ft}\) and whose altitude is \(15.3 \mathrm{ft}\).

Step-by-Step Solution

Answer

The areas are (a) 13.334 ft², (b) 368.58 m², (c) 164.72 cm², (d) 102,296 in², (e) 5.90 m², (f) 446.295 ft².

Step 1: Calculate Area of Parallelogram (a)

The area of a parallelogram is given by the formula: Area = base * altitude. For parallelogram (a), base = 4.52 ft and altitude = 2.95 ft. So, its area is calculated as: Area = 4.52 ft * 2.95 ft.

Step 2: Calculate Area of Parallelogram (b)

Using the same formula for parallelogram (b), base = 16.3 m and altitude = 22.6 m. The area is calculated as: Area = 16.3 m * 22.6 m.

Step 3: Calculate Area of Rhombus (c)

The area of a rhombus is also calculated using the base times the altitude. For rhombus (c), base = 14.2 cm and altitude = 11.6 cm. Therefore, Area = 14.2 cm * 11.6 cm.

Step 4: Calculate Area of Rhombus (d)

Like the previous rhombus, for rhombus (d) with base = 382 in and altitude = 268 in, the area is given by: Area = 382 in * 268 in.

Step 5: Calculate Area of Trapezoid (e)

The formula for the area of a trapezoid is: Area = (base1 + base2) / 2 * altitude. For trapezoid (e), the bases are 3.83 m and 2.44 m, and the altitude is 1.86 m. So, the area is: Area = (3.83 m + 2.44 m) / 2 * 1.86 m.

Step 6: Calculate Area of Trapezoid (f)

Using the trapezoid area formula for (f), with bases 33.6 ft and 24.7 ft and altitude 15.3 ft, we find the area: Area = (33.6 ft + 24.7 ft) / 2 * 15.3 ft.

Key Concepts

Parallelogram Area Calculation

Understanding the area of a parallelogram is straightforward when you know its base and altitude. The base refers to any of the sides, while the altitude is the perpendicular distance from the base to the opposite side. To calculate the area, simply multiply the length of the base by the altitude.

For example, if a parallelogram has a base of 4.52 feet and an altitude of 2.95 feet, its area is found by multiplying these two numbers.
Using the formula:
\[ \text{Area} = \text{base} \times \text{altitude} \]
we get:
\[ \text{Area} = 4.52 \, \text{ft} \times 2.95 \, \text{ft} = 13.334 \, \text{ft}^2 \].

This formula is always applicable for parallelograms, making it a powerful tool in geometry.

Rhombus Area Calculation

A rhombus, though often misunderstood, is actually a type of parallelogram with all four sides of equal length. For calculating the area of a rhombus, we also use the formula for the area of a parallelogram, because, fundamentally, they share the same properties regarding their opposite sides being parallel and equal in length.

When working with the area of a rhombus, it's pivotal to know the base and the perpendicular height (altitude) from the base to the opposite side. So the area is the product of these measurements. As a reminder, the formula is:
\[ \text{Area} = \text{base} \times \text{altitude} \].

Applying the formula to a rhombus with a base of 14.2 cm and an altitude of 11.6 cm, we find that:
\[ \text{Area} = 14.2 \, \text{cm} \times 11.6 \, \text{cm} = 164.72 \, \text{cm}^2 \].

This calculation simplifies the process and allows you to quickly determine the area of any rhombus, once the base and altitude are known.

Trapezoid Area Formula

A trapezoid, also known as a trapezium in some countries, is a four-sided figure with at least one pair of parallel sides, called the bases. The area of a trapezoid can be a bit trickier to calculate than that of a parallelogram or a rhombus, but there's a formula to make it easy.

The trapezoid area formula is:
\[ \text{Area} = \frac{(\text{base}_1 + \text{base}_2)}{2} \times \text{altitude} \].

The bases are the two parallel sides, and the altitude is the distance between them. To find the area of a trapezoid with bases of 3.83 m and 2.44 m, and an altitude of 1.86 m, we simply plug these values into the formula:
\[ \text{Area} = \frac{(3.83 \, \text{m} + 2.44 \, \text{m})}{2} \times 1.86 \, \text{m} = 5.8911 \, \text{m}^2 \].

Using this formula, irrespective of the trapezoid's shape, allows for a rapid determination of its area.