Problem 32 - Precalculus Mathematics for Calculus

Question

Find the area of the triangle whose sides have the given lengths. \(a=11, \quad b=100, \quad c=101\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 550 square units.

Step 1: Verify if the triangle is valid

To confirm that the given side lengths can form a triangle, check if the sum of the lengths of any two sides is greater than the third side. Check the inequalities: \( a + b > c \), \( a + c > b \), and \( b + c > a \). For the sides 11, 100, and 101: \(11 + 100 > 101\), \(11 + 101 > 100\), and \(100 + 101 > 11\). All inequalities hold true, confirming that the side lengths can form a triangle.

Step 2: Calculate the semi-perimeter

The semi-perimeter \( s \) of a triangle is calculated as \( \frac{a + b + c}{2} \). Substitute the given side lengths: \( s = \frac{11 + 100 + 101}{2} = 106 \).

Step 3: Apply Heron's formula

Heron's formula allows us to find the area of a triangle based on its side lengths. The area \( A \) is given by \( \sqrt{s(s-a)(s-b)(s-c)} \). Compute the components: \( s-a = 106-11 = 95 \), \( s-b = 106-100 = 6 \), and \( s-c = 106-101 = 5 \). Then, substitute these into the formula: \( A = \sqrt{106 \times 95 \times 6 \times 5} \).

Step 4: Calculate the area

Continue to compute \( A = \sqrt{106 \times 95 \times 6 \times 5} \), which equals \( A = \sqrt{302700} \). Compute the square root to find \( A \approx 550 \). The area of the triangle is approximately 550 square units.

Key Concepts

Triangle Inequalities

Before you can calculate the area of a triangle using Heron's formula or any other method, it is essential to verify that the given side lengths can indeed form a triangle. This verification comes from the concept of triangle inequalities. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

For example, if we have a triangle with sides labeled as \(a\), \(b\), and \(c\), the inequalities are as follows:

• \(a + b > c\)
• \(a + c > b\)
• \(b + c > a\)

In our case, with side lengths 11, 100, and 101:

• \(11 + 100 = 111 > 101\)
• \(11 + 101 = 112 > 100\)
• \(100 + 101 = 201 > 11\)

All these conditions are satisfied, confirming that a triangle with these sides is possible.

Semi-Perimeter

Once you've confirmed that the side lengths can form a triangle, the next step in using Heron's formula is to calculate the semi-perimeter (often denoted as \(s\)). The semi-perimeter is simply half of the triangle's perimeter.The formula for the semi-perimeter \(s\) is:
\[ s = \frac{a + b + c}{2} \]In our triangle with sides 11, 100, and 101:

• Add the side lengths: \(11 + 100 + 101 = 212\)
• Divide by 2: \(s = \frac{212}{2} = 106\)

The semi-perimeter \(s\) is 106.

This value is crucial as it is used as part of Heron's formula to find the area of the triangle.

Area Calculation

Heron's formula is a powerful tool for calculating the area of a triangle when you know the lengths of all three sides. Once you have the semi-perimeter \(s\), you can apply Heron's formula to calculate the area \(A\).The formula is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Where \(s\) is the semi-perimeter, and \(a\), \(b\), and \(c\) are the side lengths of the triangle.

• First, calculate each of the differences: \(s-a = 95\), \(s-b = 6\), and \(s-c = 5\)
• Then substitute these values into the formula: \( A = \sqrt{106 \times 95 \times 6 \times 5}\)

After calculating the product within the square root and then the square root itself, you will find:\( A \approx 550 \).The area of the triangle is approximately 550 square units.

Precalculus

Precalculus is a mathematical course that prepares students for calculus. It involves reviewing and expanding on concepts learned in algebra and geometry. Heron's formula and the concepts it involves, such as the semi-perimeter and triangle inequalities, are significant parts of precalculus.

In precalculus, students learn to combine different mathematical concepts to solve complex problems, like finding the area of a triangle just from its side lengths. This course develops critical problem-solving skills and lays the foundation for calculus, which delves deeper into changes and motion.

• Understanding these geometric principles helps solidify the foundational knowledge required for more advanced topics.
• Equips students with techniques that are applicable in various scientific and engineering fields.

The practice of verifying inequalities and calculating semi-perimeters naturally leads into calculus concepts, such as limits and derivatives, by establishing a firm grasp on mathematical relationships.