Question

In rectangle $ABCD$, the area of the shaded region is given by $\frac{\pi lw}{4}$. If the area of the shaded region is $7\pi$, what is the total area, to the nearest whole number, of the unshaded regions of rectangular $ABCD$?

• A. $4$
• B. $6$
• C. $8$
• D. $9$
• E. $10$

Answer 1

The correct option is B $6$

From the given figure,

Area of the rectangle $ABCD$ $=$ $Length$ $\times$ $Width$

$=$ $BC$ $\times$ $AB$

$=$ $l$ $\times$ $w$

$=$ $lw$

Given, area of the shaded region is given by $\frac{\pi lw}{4}$.

Area of the shaded region $=$ $7\pi$

$\Rightarrow \frac{\pi lw}{4}$ $=$ $7\pi$

We need ${}^{\prime }l{w}^{\prime }$ in ${}^{\prime }LH{S}^{\prime }$. Rearranging the terms, we get

$lw$ $=$ $\frac{7\pi \times 4}{\pi }$

$\Rightarrow lw$ $=$ $7$ $\times$ $4$

$\Rightarrow lw=$ $28$

$\Rightarrow lw$ $=$ $28$

To find total area of the unshaded regions,

From the given figure, total area of the unshaded regions $=$ (Area of the rectangle $ABCD$) $-$ (Area of the shaded region)

$=$ $\left(lw\right)$ $-$ $7\pi$

$=$ $28$ $-$ $7\pi$

$=$ $28$ $-$ $22.001$

$=$ $5.999$

$=$ $6$

Therefore, Total area of the unshaded regions $=$ ${}^{\prime }{6}^{\prime }$.