Final answer:
To represent the shaded area of a rectangle in an expression, multiply its dimensions. When comparing similar shapes with different sizes, use the squares of their scaling factors. For estimating areas under curves, set the rectangle's height at the curve's peak and adjust the width accordingly.
Step-by-step explanation:
To find an expression that represents the shaded area of a rectangle, one must identify the rectangle's length and width, multiply these two dimensions to find the area, and apply any additional steps necessary for the shaded portion. If, for example, only part of the rectangle is shaded, you may need to subtract the unshaded areas from the total area of the rectangle. Additionally, if there's a shaded region within another shape, like a square with a circle inside, the process may involve finding the area of both shapes and then subtracting the area of the shape that is not shaded from the area of the shape that is shaded.
In more complex situations like integrating under a curve to find displacement or comparing the area of two squares where the dimensions of one square are multiples of the other, the process will include establishing the relationship between the shapes or using concepts from calculus. When comparing areas of similar shapes with different dimensions, the areas are proportional to the square of the scaling factor between the dimensions. For example, if Marta's larger square has sides twice as long as her smaller square, the area of the larger square will be four times the area of the smaller square since the side lengths are squared to find the area (Area = side length2).
LibreTexts suggest that when estimating the area under a curve, one may create a rectangle with a similar area for simplicity. This requires identifying the height of the rectangle as the peak of the curve and determining the width that yields an equivalent area.
Remember, when computing areas and converting units, apply the correct number of significant figures, especially when using conversion factors like converting centimeters to meters. Exact numbers do not affect the total count of significant figures, as seen in LibreTexts