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Calculate the area of the rectangle when $a=4$ and $b=2$ . Next, calculate the area when the length and width are doubled. Does the area double? What is the ratio of the two areas you calculated? $\newline$ (A) NO; $1:4$ $\newline$ (B) YES; $1:4$ $\newline$ (C) NO; $1:2$ $\newline$ (D) YES; $1:2$

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Area of quadrilaterals and triangles: word problems

Full solution

Q. Calculate the area of the rectangle when

a=4

and

b=2

. Next, calculate the area when the length and width are doubled. Does the area double? What is the ratio of the two areas you calculated?

\newline

(A) NO;

1:4

\newline

(B) YES;

1:4

\newline

(C) NO;

1:2

\newline

(D) YES;

1:2

Calculate Area: Calculate the area of the rectangle with the given dimensions. $\newline$ The area of a rectangle is calculated by multiplying its length by its width. Given that the length $a$ is $4\,\text{cm}$ and the width $b$ is $2\,\text{cm}$ , the area $A$ can be calculated as follows: $\newline$ $A = a \times b$ $\newline$ $A = 4\,\text{cm} \times 2\,\text{cm}$ $\newline$ $A = 8\,\text{cm}^2$
Double Dimensions: Calculate the area of the rectangle when both the length and the width are doubled. $\newline$ Doubling the length and width means the new length is $2 \times 4 \, \text{cm} = 8 \, \text{cm}$ and the new width is $2 \times 2 \, \text{cm} = 4 \, \text{cm}$ . The new area $(A')$ is: $\newline$ $A' = (2 \times a) \times (2 \times b)$ $\newline$ $A' = 8 \, \text{cm} \times 4 \, \text{cm}$ $\newline$ $A' = 32 \, \text{cm}^2$
Check Doubling: Determine if the area doubles when the length and width are doubled. $\newline$ To check if the area doubles, we compare the new area ( $A'$ ) to the original area ( $A$ ). If $A'$ is exactly twice $A$ , then the area has doubled. $\newline$ $A' = 32 \, \text{cm}^2$ $\newline$ $A = 8 \, \text{cm}^2$ $\newline$ Does $A' = 2 \times A$ ? $\newline$ $32 \, \text{cm}^2 \neq 2 \times 8 \, \text{cm}^2$ (which would be $16 \, \text{cm}^2$ ) $\newline$ Therefore, the area does not double; it quadruples.
Calculate Ratio: Calculate the ratio of the new area to the original area. $\newline$ The ratio of the new area ( $A'$ ) to the original area ( $A$ ) is: $\newline$ Ratio = $\frac{A'}{A}$ $\newline$ Ratio = $\frac{32\,\text{cm}^2}{8\,\text{cm}^2}$ $\newline$ Ratio = $\frac{4}{1}$ $\newline$ The ratio of the two areas is $4:1$ .

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