Calculate the area of a rectangle with sides 3007m and 2993m, and evaluate 250092 and 2032 using algebraic…

Question

Calculate the area of a rectangle with sides 3007m and 2993m, and evaluate 250092 and 2032 using algebraic identities.

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems:

a) To find the area of a rectangular piece of land with dimensions $3007\,\text{m}$ by $2993\,\text{m}$ , we use the identity $(a+b)(a-b) = a^2 - b^2$ .

Express the dimensions in the form $(a+b)$ and $(a-b)$ .
$$ 3007 = 3000 + 7 $$ $$ 2993 = 3000 - 7 $$ So, $a = 3000$ and $b = 7$.

$$
\text{Area} = (3000 + 7)(3000 - 7) = 3000^2 - 7^2
$$

Calculate the squares.
$$ 3000^2 = 9000000 $$ $$ 7^2 = 49 $$
Subtract the values.
$$ \text{Area} = 9000000 - 49 = 8999951 $$ The area is $\boxed{\text{8999951 m}^2}$.
2. b) To evaluate $25009^2$ , we use the identity $(a+b)^2 = a^2 + 2ab + b^2$ .
Express 25009 in the form $(a+b)$ .
$$ 25009 = 25000 + 9 $$ So, $a = 25000$ and $b = 9$.

$$
25009^2 = (25000 + 9)^2 = 25000^2 + 2(25000)(9) + 9^2
$$

$$
25000^2 = 625000000
$$
$$
2(25000)(9) = 50000 \times 9 = 450000
$$
$$
9^2 = 81
$$

Add the terms.
$$ 25009^2 = 625000000 + 450000 + 81 = 625450081 $$ The value is $\boxed{625450081}$.
3. c) To find the area of a square marrow with dimensions $203\,\text{cm}$ , we use the identity $(a+b)^2 = a^2 + 2ab + b^2$ .
Express 203 in the form $(a+b)$ .
$$ 203 = 200 + 3 $$ So, $a = 200$ and $b = 3$.

$$
\text{Area} = 203^2 = (200 + 3)^2 = 200^2 + 2(200)(3) + 3^2
$$

$$
200^2 = 40000
$$
$$
2(200)(3) = 400 \times 3 = 1200
$$
$$
3^2 = 9
$$

$$
\text{Area} = 40000 + 1200 + 9 = 41209
$$
The area is $\boxed{\text{41209 cm}^2}$.

Calculate the area of a rectangle with sides 3007m and 2993m, and evaluate 250092 and 2032 using algebraic identities.