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.