### Understanding the Area of EllipsesAn ellipse is an oval-shaped closed curve. The area of an ellipse is approximately $3.14 \times a \times b$, where $a$ is its length and $b$ is its width. Let's find the polynomial that approximates the total area of both elliptical-shaped lenses of the sunglasses shown in the diagram. **Assumption:** Assume $c = 1$. Simplify your answer completely.![Sunglasses Diagram](image-url)Given data from the diagram:- Length of each lens (major axis): $(x + c)$ inches- Width of each lens (minor axis): $(x - c)$ inches### Steps to Calculate the Area1. **Calculate the Area of One Lens:** The area $A$ of an ellipse can be calculated using the formula: \[ A = 3.14 \times a \times b \] For one lens, \[ A_{\text{one lens}} = 3.14 \times (x + c) \times (x - c) \]2. **Simplify the Area Formula:** Using the difference of squares identity $(a + b)(a - b) = a^2 - b^2$, \[ A_{\text{one lens}} = 3.14 \times (x^2 - c^2) \] Given $c = 1$, \[ A_{\text{one lens}} = 3.14 (x^2 - 1^2) = 3.14 (x^2 - 1) \]3. **Calculate the Total Area for Both Lenses:** Since there are two lenses, \[ A_{\text{total}} = 2 \times A_{\text{one lens}} = 2 \times 3.14 (x^2 - 1) \] Simplifying this, \[ A_{\text{total}} = 6.28 (x^2 - 1) \quad \text{square inches} \]### ConclusionThe polynomial that approximates the total area of both elliptical-shaped lenses is:\[6.28 (x^2 - 1)\]

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An ellipse is an oval-shaped closed curve. The area of an ellipse is approximately 3.14ab, where a is its length and b is its width. Find the polynomial that approximates the total area of both elliptical- shaped lenses of the sunglasses shown. (Assume c = 1. Simplify your answer completely.) (x– C) in. (x + c) in.

An ellipse is an oval-shaped closed curve. The area of an ellipse is approximately 3.14ab, where a is its length and b is its width. Find the polynomial that approximates the total area of both elliptical- shaped lenses of the sunglasses shown. (Assume c = 1. Simplify your answer completely.) (x– C) in. (x + c) in.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Concept explainers

Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.

Question

### Understanding the Area of Ellipses

An ellipse is an oval-shaped closed curve. The area of an ellipse is approximately $3.14 \times a \times b$, where $a$ is its length and $b$ is its width. Let's find the polynomial that approximates the total area of both elliptical-shaped lenses of the sunglasses shown in the diagram.

**Assumption:** Assume $c = 1$. Simplify your answer completely.

![Sunglasses Diagram](image-url)

Given data from the diagram:
- Length of each lens (major axis): $(x + c)$ inches
- Width of each lens (minor axis): $(x - c)$ inches

### Steps to Calculate the Area

1. **Calculate the Area of One Lens:**

The area $A$ of an ellipse can be calculated using the formula:
\[
A = 3.14 \times a \times b
\]
For one lens,
\[
A_{\text{one lens}} = 3.14 \times (x + c) \times (x - c)
\]

2. **Simplify the Area Formula:**

Using the difference of squares identity $(a + b)(a - b) = a^2 - b^2$,
\[
A_{\text{one lens}} = 3.14 \times (x^2 - c^2)
\]
Given $c = 1$,
\[
A_{\text{one lens}} = 3.14 (x^2 - 1^2) = 3.14 (x^2 - 1)
\]

3. **Calculate the Total Area for Both Lenses:**

Since there are two lenses,
\[
A_{\text{total}} = 2 \times A_{\text{one lens}} = 2 \times 3.14 (x^2 - 1)
\]
Simplifying this,
\[
A_{\text{total}} = 6.28 (x^2 - 1) \quad \text{square inches}
\]

### Conclusion

The polynomial that approximates the total area of both elliptical-shaped lenses is:
\[
6.28 (x^2 - 1)
\]

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