Question

given that a rectangle has a length of $\frac{5}{2}x + 10$ with a width of $\frac{5}{2}x + 5$, formulate an expression to represents the area of the rectangle.
a $\frac{25x^{2}}{2}+\frac{75x}{2}+50$
b $\frac{25x^{2}}{4}+\frac{75x}{4}+50$
c $\frac{25x^{2}}{4}+\frac{75x}{2}+50$
d $\frac{25x^{2}}{2}+\frac{75x}{4}+50$
e $\frac{25x^{2}}{3}+\frac{75x}{4}+50$

Explanation:

Step1: Recall area formula

The area $A$ of a rectangle is $A = \text{length}\times\text{width}$. Here, length $l=\frac{5}{2}x + 10$ and width $w=\frac{5}{2}x + 5$.

Step2: Multiply the expressions

$A=(\frac{5}{2}x + 10)(\frac{5}{2}x + 5)$. Using the FOIL - method:
First terms: $\frac{5}{2}x\times\frac{5}{2}x=\frac{25x^{2}}{4}$.
Outer terms: $\frac{5}{2}x\times5=\frac{25x}{2}$.
Inner terms: $10\times\frac{5}{2}x = 25x$.
Last terms: $10\times5 = 50$.

Step3: Combine like - terms

$A=\frac{25x^{2}}{4}+(\frac{25x}{2}+25x)+50=\frac{25x^{2}}{4}+(\frac{25x + 50x}{2})+50=\frac{25x^{2}}{4}+\frac{75x}{2}+50$.

Answer:

C. $\frac{25x^{2}}{4}+\frac{75x}{2}+50$