Tabitha wants to hang a painting in a gallery. The painting and frame must have an area of 58 square feet. The painting is 7 feet wide by 8 feet long. Which quadratic equation can be used to determine the thickness of the frame, x?

To solve the problem of finding the thickness of the frame \( x \), we need to set up an equation based on the given conditions.

### Step 1: Determine the area of the painting
The painting is 7 feet wide and 8 feet long, so its area is:
\[
\text{Area of the painting} = 7 \times 8 = 56 \text{ square feet}
\]

### Step 2: Express the total area of the painting and frame together
The frame adds a thickness of \( x \) feet on all sides of the painting. Therefore, the dimensions of the painting including the frame will be:
- Width: \( 7 + 2x \) (since \( x \) is added to both sides)
- Length: \( 8 + 2x \)

So, the total area of the painting and frame combined is:
\[
\text{Total area} = (7 + 2x)(8 + 2x)
\]

### Step 3: Set up the quadratic equation
We know the total area (including the frame) should be 58 square feet. Therefore:
\[
(7 + 2x)(8 + 2x) = 58
\]

### Step 4: Expand the equation
Expand the product on the left side:
\[
(7 + 2x)(8 + 2x) = 7 \times 8 + 7 \times 2x + 8 \times 2x + 2x \times 2x
\]
\[
= 56 + 14x + 16x + 4x^2
\]
\[
= 4x^2 + 30x + 56
\]

### Step 5: Form the final quadratic equation
Now, set the expanded expression equal to the total area of 58 square feet:
\[
4x^2 + 30x + 56 = 58
\]

Finally, subtract 58 from both sides to form the quadratic equation:
\[
4x^2 + 30x - 2 = 0
\]

So, the quadratic equation that can be used to determine the thickness of the frame \( x \) is:
\[
4x^2 + 30x - 2 = 0
\]

Would you like a detailed explanation of how to solve this quadratic equation or have any questions?

Here are 8 related questions to expand your understanding:
1. How would you solve the quadratic equation \(4x^2 + 30x - 2 = 0\)?
2. What does the discriminant of this quadratic equation indicate about the number of solutions?
3. How would the equation change if the painting had different dimensions?
4. Can you derive a general formula for the frame thickness given any painting dimensions and total area?
5. What would happen to the thickness \( x \) if the total area were smaller than the painting area?
6. How does the thickness \( x \) change if the total area increases?
7. Could you solve the quadratic equation graphically?
8. How would you check the correctness of your solution?

**Tip:** When expanding and simplifying algebraic expressions, ensure each term is carefully accounted for to avoid errors.

Learn how to determine the thickness of a frame using a quadratic equation. Step-by-step solution provided for solving 4x^2 + 30x - 2 = 0, given the area requirements of a painting and frame totaling 58 square feet. Suitable for high school students studying algebra and quadratic equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation