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?

We are given that the area of the rectangle is equal to the area of the triangle, and we need to find the value of \( x \).

### Step 1: Calculate the area of the rectangle

The formula for the area of a rectangle is:

\[
\text{Area of rectangle} = \text{length} \times \text{width}
\]

Given:
- Length of the rectangle = \( 5 + 2x \) inches
- Width of the rectangle = 10 inches

\[
\text{Area of rectangle} = (5 + 2x) \times 10 = 50 + 20x
\]

### Step 2: Calculate the area of the triangle

The formula for the area of a triangle is:

\[
\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]

Given:
- Base of the triangle = 30 inches
- Height of the triangle = \( 4x - 10 \) inches

\[
\text{Area of triangle} = \frac{1}{2} \times 30 \times (4x - 10) = 15 \times (4x - 10) = 60x - 150
\]

### Step 3: Set up the equation

Since the areas are equal, we equate the two expressions:

\[
50 + 20x = 60x - 150
\]

### Step 4: Solve for \( x \)

First, subtract \( 20x \) from both sides:

\[
50 = 40x - 150
\]

Next, add 150 to both sides:

\[
200 = 40x
\]

Now, divide both sides by 40:

\[
x = \frac{200}{40} = 5
\]

### Step 5: Conclusion

The value of \( x \) is \( \boxed{5} \).

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:
1. What happens to the area of the rectangle as \( x \) increases?
2. How would you solve if the width of the rectangle was also variable?
3. Can you find the perimeter of the rectangle for \( x = 5 \)?
4. What is the area of the triangle when \( x = 5 \)?
5. How would the equation change if the triangle had a different base?

**Tip**: Always verify your solution by plugging the value of \( x \) back into the original expressions to check for consistency!

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?

Learn how to find the value of x when the area of a rectangle is equal to the area of a triangle. The rectangle has a length of (5 + 2x) inches and width of 10 inches, while the triangle has a base of 30 inches and height of (4x - 10) inches. This step-by-step solution involves key algebraic and geometric concepts, suitable for students in Grades 7-9.

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