https://file.aiquickdraw.com/mathbot/file/1727867290902ahayma8a.jpg

We are given the dimensions of a rectangle as \( 0.7x \) and \( 5 - 3x \). We are asked to find the value of \( x \) that maximizes the area of the rectangle and the corresponding maximum area.

### Step 1: Expression for the area
The area \( A \) of the rectangle is given by the product of its length and width:
\[
A(x) = (0.7x)(5 - 3x)
\]
Simplifying this expression:
\[
A(x) = 0.7x(5 - 3x) = 3.5x - 2.1x^2
\]

### Step 2: Find the value of \( x \) that maximizes the area
To maximize the area, we take the derivative of \( A(x) \) with respect to \( x \) and set it to zero:
\[
\frac{dA}{dx} = 3.5 - 4.2x
\]
Setting \( \frac{dA}{dx} = 0 \) to find the critical points:
\[
3.5 - 4.2x = 0
\]
\[
4.2x = 3.5
\]
\[
x = \frac{3.5}{4.2} = 0.8333
\]

### Step 3: Check if it is a maximum
To confirm that this value gives a maximum, we check the second derivative:
\[
\frac{d^2A}{dx^2} = -4.2
\]
Since the second derivative is negative, \( x = 0.83 \) gives a maximum.

### Step 4: Calculate the maximum area
Now substitute \( x = 0.83 \) into the area equation:
\[
A(0.83) = 3.5(0.83) - 2.1(0.83)^2
\]
\[
A(0.83) = 2.905 - 2.1(0.6889) = 2.905 - 1.44669 = 1.4583 \, \text{square units}
\]

Thus, the value of \( x \) that gives the maximum area is approximately \( x = 0.83 \), and the maximum area is approximately \( 1.46 \, \text{square units} \).

### Conclusion
The correct answer is:
- **Option 1: When \( x = 0.83 \), the maximum area is 1.46 square units.**

Would you like further details or have any questions?

---

Here are 5 related questions:
1. How would the area change if the dimensions of the rectangle were altered?
2. What happens if we try to minimize the area instead of maximizing it?
3. How do you find the critical points of a more complex function?
4. Why does the second derivative help in determining whether a point is a maximum or minimum?
5. How can we visualize the area function and its maximum point?

**Tip:** For problems involving maximization or minimization, always check the second derivative to confirm if a critical point gives a maximum or a minimum.

A rectangle has dimensions 0.7x and 5 - 3x. What value of x gives the maximum area and what is the maximum area?

Learn how to find the value of x that maximizes the area of a rectangle with dimensions 0.7x and 5 - 3x. This involves algebra, the quadratic equation, and using derivatives for optimization. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation