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.


We are given the following information:

1. **Rectangle Dimensions:**
   - Length = \(5 + 2x\) inches
   - Width = 10 inches

2. **Triangle Dimensions:**
   - Base = 30 inches
   - Height = \(4x - 10\) inches

### Step 1: Write the expressions for the areas

- The area \(A_{\text{rectangle}}\) of a rectangle is given by:
\[
A_{\text{rectangle}} = \text{Length} \times \text{Width} = (5 + 2x) \times 10
\]
So,
\[
A_{\text{rectangle}} = 10(5 + 2x) = 50 + 20x
\]

- The area \(A_{\text{triangle}}\) of a triangle is given by:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 30 \times (4x - 10)
\]
Simplifying,
\[
A_{\text{triangle}} = 15(4x - 10) = 60x - 150
\]

### Step 2: 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 3: Solve for \(x\)

1. Subtract \(20x\) from both sides:
\[
50 = 40x - 150
\]

2. Add 150 to both sides:
\[
200 = 40x
\]

3. Divide by 40:
\[
x = 5
\]

### Step 4: Check the result

- For \(x = 5\):
  - Rectangle: \(5 + 2(5) = 5 + 10 = 15\) inches (length).
  - Area of the rectangle = \(15 \times 10 = 150\) square inches.
  
  - Triangle: \(4(5) - 10 = 20 - 10 = 10\) inches (height).
  - Area of the triangle = \(\frac{1}{2} \times 30 \times 10 = 150\) square inches.

Since both areas are 150 square inches, the solution is correct.

### Final Answer: \(x = 5\).

Would you like further clarification on any of the steps?

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Here are 5 related questions you might consider:

1. How do we derive the formula for the area of a rectangle?
2. Can we solve for the dimensions of the triangle using the found value of \(x\)?
3. What would happen if the rectangle and triangle had different areas?
4. How can we solve similar problems with other geometric shapes?
5. How can we express the area of a rectangle and triangle using algebraic terms?

**Tip:** Always remember to verify your solution by substituting the values back into the original problem.

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.

In this problem, we solve for x by equating the area of a rectangle with dimensions (5 + 2x) inches by 10 inches and a triangle with base 30 inches and height (4x - 10) inches. This exercise involves algebraic manipulation and geometry concepts, ideal for Grades 9-11.

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