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.

What is the value of x?

To find the value of \( x \), we will use the information given about the areas of both the rectangle and the triangle. We are told that the area of the rectangle is equal to the area of the triangle.

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

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

### Step 3: Set up the equation
We are told that the area of the rectangle is equal to the area of the triangle. Therefore, we can set up the equation:
\[
50 + 20x = 60x - 150
\]

### Step 4: Solve for \( x \)
First, subtract \( 20x \) from both sides to get all terms involving \( x \) on one side:
\[
50 = 40x - 150
\]
Now, add 150 to both sides:
\[
200 = 40x
\]
Finally, divide both sides by 40:
\[
x = \frac{200}{40} = 5
\]

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

Would you like further details on any part of this explanation? Here are some related questions to explore:

1. What is the final area of both the rectangle and triangle when \( x = 5 \)?
2. How does the length of the rectangle change as \( x \) increases?
3. What happens to the height of the triangle if \( x \) is negative?
4. Can the value of \( x \) ever result in a negative area for either shape?
5. How do different values of \( x \) affect the proportionality of the rectangle's and triangle's areas?

**Tip**: When solving problems with equations, it's useful to ensure units are consistent throughout.

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. What is the value of x?

Find the value of x in a problem where the areas of a rectangle and a triangle are equal. This problem involves algebraic equations, rectangle area, triangle area, and solving for variables. Suitable for students in Grades 8-10.

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