A rectangle has a width of 3 centimeters. If the length is increased by 6 centimeters, by how much will the area, in square centimeters, increase?
The area will increase by 18 square centimeters.
The area of a rectangle is calculated using the formula A = length × width. Initially, the rectangle has a width of 3 cm and a certain length, which we can denote as L. When the length is increased by 6 cm, the new length becomes L + 6 cm, resulting in the new area being (L + 6) × 3 cm. The increase in area can be calculated by subtracting the original area from the new area.
This choice suggests a minimal increase in area, which is incorrect. The original area is 3L, and the new area becomes 3(L + 6). The difference between these areas is 3L + 18 - 3L = 18, not 3. Thus, this option underestimates the increase in area.
Choosing 6 as the increase implies that the additional area gained from the lengthening is minimal. However, as previously calculated, the area increases by 18 square centimeters, making this answer incorrect. The increase due to the additional 6 cm in length contributes significantly more to the overall area.
This option represents another miscalculation of the area increase. While it may seem plausible to assume a linear growth, the actual increase results from a product of the increased length and the constant width. The calculation shows that the correct increase is 18 square centimeters, not 12.
This is the correct choice because the area increase is calculated as 3 cm (width) multiplied by the increase in length (6 cm), which equals 18 square centimeters. This accurately reflects the total increase in area resulting from the increased length of the rectangle.
The increase in area of a rectangle is directly related to both the width and the change in length. In this case, increasing the length by 6 cm while keeping the width constant at 3 cm leads to a total area increase of 18 square centimeters, confirming that D is the correct answer. Understanding how to apply the area formula is essential in solving problems involving geometric shapes.
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