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?
A.2.5
B.3
C.7.5
D.5

To solve for the value of \( x \), we start by calculating the areas of the rectangle and the triangle and then set them equal to each other.

1. **Area of the Rectangle:**

The formula for the area of a rectangle is:
\[
\text{Area} = \text{Length} \times \text{Width}
\]

Given:
\[
\text{Length} = 5 + 2x \quad \text{inches}
\]
\[
\text{Width} = 10 \quad \text{inches}
\]

So, the area of the rectangle is:
\[
\text{Area}_{\text{Rectangle}} = (5 + 2x) \times 10 = 50 + 20x
\]

2. **Area of the Triangle:**

The formula for the area of a triangle is:
\[
\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}
\]

Given:
\[
\text{Base} = 30 \quad \text{inches}
\]
\[
\text{Height} = 4x - 10 \quad \text{inches}
\]

So, the area of the triangle is:
\[
\text{Area}_{\text{Triangle}} = \frac{1}{2} \times 30 \times (4x - 10)
\]
Simplifying further:
\[
\text{Area}_{\text{Triangle}} = 15 \times (4x - 10) = 60x - 150
\]

3. **Setting the Areas Equal:**

We set the area of the rectangle equal to the area of the triangle:
\[
50 + 20x = 60x - 150
\]

4. **Solving for \( x \):**

Rearrange the equation to isolate \( x \):
\[
50 + 150 = 60x - 20x
\]
\[
200 = 40x
\]

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

Therefore, the value of \( x \) is:

**D. 5**

Would you like a detailed explanation or have any questions about the solution? 

Here are some related questions for further practice:

1. If the length of the rectangle is doubled, what would the new area be?
2. What is the value of \( x \) if the triangle's base was 25 inches?
3. Find the perimeter of the rectangle when \( x = 5 \).
4. What would the height of the triangle be if \( x = 3 \)?
5. How would the area of the rectangle change if the width was reduced by 2 inches?

**Tip:** When equating the areas of different shapes, ensure that the units and dimensions correspond to avoid miscalculations.

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? A. 2.5 B. 3 C. 7.5 D. 5

Learn how to solve for x in a problem where the area of a rectangle is equal to the area of a triangle. Using algebra and geometry, this step-by-step solution demonstrates the key formulas for area calculation. Suitable for students in Grades 8-10.

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