Expressing the Trapezoid Area: Using X to Account for Base Differences

To solve this problem, we will find the area of the trapezoid using the given expressions for the bases and height.

Step 1: Determine the height of the trapezoid.

• The height, $h$, is given as the sum of the two bases: $h = X + (X + 5) = 2X + 5$

Step 2: Apply the formula for the area of a trapezoid.

• The formula for the area of a trapezoid is: $\text{Area} = \frac{1}{2} \times (\text{small base} + \text{large base}) \times \text{height}$
• Substitute the expressions for the bases and height: $\text{Area} = \frac{1}{2} \times (X + (X + 5)) \times (2X + 5)$
• Further simplifying: $\text{Area} = \frac{1}{2} \times (2X + 5) \times (2X + 5)$
• This becomes: $\text{Area} = \frac{1}{2} \times (2X + 5)^2$
• Expand $(2X + 5)^2$ using the square of a binomial formula: $(2X + 5)^2 = 4X^2 + 20X + 25$
• Thus the area simplifies to: $\text{Area} = \frac{1}{2} (4X^2 + 20X + 25)$

Therefore, the expression for the area of the trapezoid in terms of $X$ is $\frac{1}{2}(4X^2 + 20X + 25)$.