Question

given that a rectangle has a length of $\frac{5}{3}x + 10$ with a width of $\frac{3}{2}x + 5$, formulate an expression to represents the area of the rectangle.
a $\frac{15x^{2}}{6}+\frac{75x}{6}+50$
b $\frac{15x^{2}}{6}+\frac{75x}{4}+50$
c $\frac{15x^{2}}{6}+\frac{75x}{2}+50$
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 formula for a rectangle is $A = l\times w$, where $l$ is the length and $w$ is the width. Here, $l=\frac{5}{3}x + 10$ and $w=\frac{3}{2}x + 5$.

Step2: Multiply the length and width

\[

$$\begin{align*} A&=(\frac{5}{3}x + 10)(\frac{3}{2}x + 5)\\ &=\frac{5}{3}x\times\frac{3}{2}x+\frac{5}{3}x\times5+10\times\frac{3}{2}x + 10\times5\\ &=\frac{15}{6}x^{2}+\frac{25}{3}x + 15x+50\\ &=\frac{5}{2}x^{2}+\frac{25 + 45}{3}x+50\\ &=\frac{5}{2}x^{2}+\frac{70}{3}x + 50 \end{align*}$$

\]

Answer:

There seems to be a formatting issue with the options provided as they are not clearly readable. But the correct expression for the area of the rectangle is $\frac{5}{2}x^{2}+\frac{70}{3}x + 50$.