Question

construction the public works department is repaving some of the roads in the city. the materials needed to repave this section of the road cost $2.50 per square foot.
a. what is the possible range for the area of the road? round the answer to the nearest hundredth if necessary.
least possible area:
ft²
greatest possible area:
ft²

Explanation:

Step1: Determine measurement precision

Assume the given dimensions (45 ft and 100 ft) are measured to the nearest whole number, so the least possible values are 44.5 ft (for 45 ft, since it could be as low as 44.5 when rounded to 45) and 99.5 ft (for 100 ft, as low as 99.5 when rounded to 100), and the greatest possible values are 45.5 ft (for 45 ft, as high as 45.5 when rounded to 45) and 100.5 ft (for 100 ft, as high as 100.5 when rounded to 100).

Step2: Calculate least possible area

The area of a rectangle is \( A = l \times w \). For the least area, use the least possible length and width: \( A_{least} = 44.5 \times 99.5 \).

$$ 44.5 \times 99.5 = (45 - 0.5)(100 - 0.5) = 45\times100 - 45\times0.5 - 0.5\times100 + 0.5\times0.5 = 4500 - 22.5 - 50 + 0.25 = 4427.75 $$

Step3: Calculate greatest possible area

For the greatest area, use the greatest possible length and width: \( A_{greatest} = 45.5 \times 100.5 \).

$$ 45.5 \times 100.5 = (45 + 0.5)(100 + 0.5) = 45\times100 + 45\times0.5 + 0.5\times100 + 0.5\times0.5 = 4500 + 22.5 + 50 + 0.25 = 4572.75 $$

Answer:

least possible area: \( 4427.75 \) ft²
greatest possible area: \( 4572.75 \) ft²