A rectangle has a length of (5+2x) inches and a width of 10 inches. A triangle has a base of 30 inches and a height of (4x-10) inches. The area in square inches of the rectangle is equal to the area in square inches of the triangle.

Let's solve this problem step by step:

### Step 1: Area of the Rectangle
The formula for the area of a rectangle is:
\[
\text{Area of rectangle} = \text{length} \times \text{width}
\]
For this rectangle, the length is \( (5 + 2x) \) inches and the width is 10 inches. So, the area of the rectangle is:
\[
\text{Area of rectangle} = (5 + 2x) \times 10 = 10(5 + 2x) = 50 + 20x \, \text{square inches}
\]

### Step 2: Area of the Triangle
The formula for the area of a triangle is:
\[
\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For this triangle, the base is 30 inches and the height is \( (4x - 10) \) inches. So, the area of the triangle is:
\[
\text{Area of triangle} = \frac{1}{2} \times 30 \times (4x - 10) = 15 \times (4x - 10) = 60x - 150 \, \text{square inches}
\]

### Step 3: Set the Areas Equal
Since the area of the rectangle is equal to the area of the triangle, we can set the two expressions equal to each other:
\[
50 + 20x = 60x - 150
\]

### Step 4: Solve for \( x \)
Now, let's solve the equation:
\[
50 + 20x = 60x - 150
\]
Subtract \( 20x \) from both sides:
\[
50 = 40x - 150
\]
Add 150 to both sides:
\[
200 = 40x
\]
Divide by 40:
\[
x = 5
\]

### Step 5: Conclusion
The value of \( x \) is 5.

Would you like further details or clarifications?

Here are some related questions to explore:
1. What would happen if the width of the rectangle was doubled?
2. How would the areas change if the height of the triangle increased by 5 inches?
3. What is the perimeter of the rectangle when \( x = 5 \)?
4. How do the areas of the two shapes compare when \( x = 0 \)?
5. Can you derive a general formula for \( x \) based on different lengths of the rectangle?

**Tip:** In geometry problems, always double-check the units to ensure consistency in your calculations.

Find the value of x when the area of a rectangle with a length of (5 + 2x) inches and width of 10 inches equals the area of a triangle with a base of 30 inches and height (4x - 10) inches. This solution involves key algebraic and geometric concepts and is suitable for Grades 8-10.

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