I am recently thinking of installing a triangular pool. Based on the measure of my backyard the sides can be a = 12.5 feet, b = 13.7 feet, and c = 20.4 feet. If the Blue Pool service is charging for $75 per square foot, how much should I expect to spend?
Round answers to the nearest hundredth.

To determine the cost of installing the triangular pool, we need to first calculate the area of the triangle using the given side lengths. We can use Heron's formula for this purpose.

Heron's formula states that the area $A$ of a triangle with sides $a$, $b$, and $c$ is given by:

\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

where $ s $ is the semi-perimeter of the triangle, calculated as:

\[ s = \frac{a + b + c}{2} \]

Given:
\[ a = 12.5 \text{ feet} \]
\[ b = 13.7 \text{ feet} \]
\[ c = 20.4 \text{ feet} \]

First, we calculate the semi-perimeter $ s $:

\[ s = \frac{12.5 + 13.7 + 20.4}{2} \]
\[ s = \frac{46.6}{2} \]
\[ s = 23.3 \text{ feet} \]

Next, we apply Heron's formula to find the area $ A $:

\[ A = \sqrt{23.3(23.3 - 12.5)(23.3 - 13.7)(23.3 - 20.4)} \]
\[ A = \sqrt{23.3 \times 10.8 \times 9.6 \times 2.9} \]
\[ A = \sqrt{6867.264} \]
\[ A \approx 82.88 \text{ square feet} \]

Now, to find the cost of installing the pool, we multiply the area by the cost per square foot:

\[ \text{Cost} = 82.88 \times 75 \]
\[ \text{Cost} = 6216.00 \]

Therefore, you should expect to spend approximately $6216.00 on the pool installation.

Would you like more details or have any questions?

Here are 5 questions you might consider asking next:
1. How do you determine if a set of side lengths can form a valid triangle?
2. Can you explain more about Heron's formula and its derivation?
3. What other methods can be used to calculate the area of a triangle?
4. How does the cost change if the cost per square foot changes?
5. What are some important factors to consider when installing a pool?

Tip: Always double-check your calculations to ensure accuracy, especially when dealing with measurements and costs.

Learn how to calculate the cost of installing a triangular pool using Heron's formula. Given side lengths a = 12.5 feet, b = 13.7 feet, and c = 20.4 feet, the area is approximately 82.88 square feet. At $75 per square foot, expect to spend approximately $6216.00. Explore step-by-step calculations and related concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation