Problem 9 - Technical Mathematics with Calculus

Question

To help you solve each problem, draw a diagram and label it completely. Look for special triangles or right triangles contained in the diagram. Be sure to look up any word that is unfamiliar. What is the cost, to the nearest dollar, of a triangular piece of land whose base is \(828 \mathrm{ft}\) and altitude is \(412 \mathrm{ft}\) at \(\$ 1125\) an acre? \(\left(1 \text { acre }=43,560 \mathrm{ft}^{2}\right).\)

Step-by-Step Solution

Answer

The cost of the triangular piece of land is $10,450 to the nearest dollar.

Step 1 - Draw and label the diagram

Draw a triangle to represent the piece of land. Label the base of the triangle as 828 ft and the altitude (height) as 412 ft. Make sure the altitude is drawn as a perpendicular line from the base to the opposite vertex.

Step 2 - Calculate the area of the triangle

Use the formula for the area of a triangle: \( A = \frac{1}{2} \times \text{base} \times \text{height} \). Substitute the base and height with the given values: \( A = \frac{1}{2} \times 828 \text{ ft} \times 412 \text{ ft} \). Calculate to find the area in square feet.

Step 3 - Convert the area to acres

Since 1 acre = 43,560 square feet, divide the area obtained in Step 2 by 43,560 to convert the area from square feet to acres.

Step 4 - Calculate the cost of the land

Multiply the area of the land in acres by the cost per acre to find the total cost. If the answer is not a whole number, round off to the nearest dollar.

Key Concepts

Area of a Triangle

Understanding how to calculate the area of a triangle is fundamental in geometry and is often applied in real-life situations, such as finding the space of a triangular land area. The area ((A)) of a triangle can be found using the formula \[ A = \frac{1}{2} \times \text{base} \times \text{height} \].

In the context of the given problem, if the base of the land is 828 ft and the altitude, which is another term for height in triangles, is 412 ft, we apply the formula by multiplying the base and height and then dividing by 2.

This results in the triangle's area in square feet, which will then be necessary to convert to acres for financial purposes, such as calculating land cost. A detailed visualization through a labeled diagram can vastly improve comprehension, enabling a clear understanding of where these numbers are coming from and how they are used within the formula.

Converting Square Feet to Acres

Once we know the area of a triangle in square feet, we often need to convert it into acres, which is a common land measure unit in the United States. There are 43,560 square feet in one acre. To convert square feet to acres, you divide the area in square feet by 43,560.

For example, if the calculated area of a triangular land plot is 170,856 square feet, we divide that number by 43,560 to convert to acres: \[ \text{Acres} = \frac{170,856 \text{ sq ft}}{43,560} \].

This conversion is particularly important when dealing with real estate, agriculture, and land development, where pricing is commonly given per acre. Remembering this conversion factor and understanding how to apply it is essential, and ensure always to carry out the division accurately to obtain a precise measure of the land.

Right Triangles

A right triangle is a triangle that has a 90-degree angle, typically denoted with a small square in diagrams. This type of triangle is significant because its side lengths are related by the Pythagorean theorem if they are all integers, and it allows for straightforward calculations of area given the base and height.

In problems regarding land calculation, such as the exercise we are discussing, we often assume that the plot of land is a right triangle for ease of measurement. This allows us to take the perpendicular distance from the base to the highest point of the triangle as the height without any complex calculations.

Special Considerations for Right Triangles

In addition to ease of area calculation, right triangles are fundamental in trigonometry, with properties that enable us to calculate unknown angles and side lengths using sine, cosine, and tangent ratios. However, for the purpose of determining the triangular land's cost, knowing the base and height is sufficient, and we apply the basic area formula that we've covered.