Question

if both dimensions are doubled, then which of the following statements is true? the new area will be 16 times the old area. the new area will be 2 times the old area. the new area will be 9 times the old area.

Explanation:

Step1: Calculate the original area

The original rectangle has length $l = 8$ in and width $w = 4$ in. The area formula for a rectangle is $A=l\times w$. So the original area $A_1=8\times4 = 32$ square - inches.

Step2: Calculate the new area when dimensions are doubled

The new length $l_2 = 2\times8=16$ in and new width $w_2 = 2\times4 = 8$ in. The new area $A_2=l_2\times w_2=16\times8 = 128$ square - inches.

Step3: Find the ratio of new area to old area

$\frac{A_2}{A_1}=\frac{128}{32}=4$. But if we consider the general rule for a rectangle, if the length $l$ and width $w$ are changed to $k_1l$ and $k_2w$ respectively, the new area $A_{new}=k_1l\times k_2w$. When $k_1 = k_2 = 2$, $A_{new}=(2l)\times(2w)=4lw$. In general, if both dimensions of a rectangle are multiplied by a factor $k$, the new area is $k^{2}$ times the old area. Here $k = 2$, so the new area is $4$ times the old area. If we double both dimensions of a rectangle, the new area will be 4 times the old area. Let's check the options:
If we assume the old area of the rectangle is $A = lw$ and the new area after doubling both dimensions is $A'=(2l)\times(2w)=4lw$.

Answer:

None of the given statements are true.