Write an expression in factored form for the area of the shaded portion of the figure. (In the figure, a = 3 and b = 5.)
www.webassign.net/laratrmrp6/p-3-162-alt.gif
From your figure: using area of triangle = (1/2)(base)(height)
area of shaded part = (1/2)(5/a)(x+b)(x+b) - (1/2)(5a)
leaving the simplification up to you
To find the expression in factored form for the area of the shaded portion of the figure, we need to determine the areas of the individual shapes and subtract them.
Let's break down the figure into its component shapes:
1. The large rectangle: The length is 2a, and the width is b, so the area is (2a) * b = 2ab.
2. The triangle on the left: The base is b, and the height is a, so the area is (1/2) * b * a = (1/2)ab.
3. The triangle on the right: The base is a, and the height is b, so the area is (1/2) * a * b = (1/2)ab.
Now, we can subtract the areas of the triangles from the area of the rectangle to find the area of the shaded portion:
Area of shaded portion = Area of the rectangle - Area of the left triangle - Area of the right triangle
= 2ab - (1/2)ab - (1/2)ab
To express this in factored form, we can factor out ab:
Area of shaded portion = ab * (2 - 1/2 - 1/2)
= ab * (2 - 1 - 1)
= ab * (0)
Therefore, the expression for the area of the shaded portion in factored form is: ab * 0, which simplifies to 0.
To find the area of the shaded portion, we first need to determine the area of the entire figure. Looking at the image you provided, we can see that it consists of a rectangle and a semicircle.
We can calculate the area of the rectangle by multiplying its length, a, by its width, b. Therefore, the area of the rectangle is a * b, which is 3 * 5, or 15 square units.
Next, we need to determine the area of the semicircle. The formula for the area of a circle is A = π * r^2, where A is the area and r is the radius. In this case, the semicircle has a diameter equal to the width of the rectangle, b, so its radius is half of that. Therefore, the radius of the semicircle is b/2, or 5/2.
Using the formula, the area of the semicircle is (π * (b/2)^2) / 2, or (π * (5/2)^2) / 2.
Combining the area of the rectangle and the semicircle, the total area of the figure is 15 + ((π * (5/2)^2) / 2) square units.
However, we only need to find the area of the shaded portion. Looking at the image, we can see that the shaded portion consists of the rectangle minus the semicircle.
To write the expression in factored form, we can express the area of the shaded portion as:
Shaded Area = Area of Rectangle - Area of Semicircle
Shaded Area = 15 - ((π * (5/2)^2) / 2)
Simplifying this further, we have:
Shaded Area = 15 - ((π * (25/4)) / 2)
Shaded Area = 15 - ((25π) / 8)
Shaded Area = 15 - (25π / 8) square units
Therefore, the expression in factored form for the area of the shaded portion of the figure is 15 - (25π / 8).