This is an image of a geometry problem where a shaded area lies within a rectangle and is bounded by part of a circle. The rectangle has dimensions $6 \, \text{units} \times 4 \, \text{units}$, and the shaded region appears to be the area of a circular segment.

We can break this problem into the following steps:

Step 1: Find the area of the rectangle

The total area of the rectangle is: $\text{Area of rectangle} = \text{length} \times \text{width} = 6 \times 4 = 24 \, \text{square units}.$

Step 2: Understand the semicircular part

The shaded area seems to be a circular segment of a circle whose diameter is equal to the width of the rectangle ($6 \, \text{units}$). The circle's radius $r$ is: $r = \frac{\text{diameter}}{2} = \frac{6}{2} = 3 \, \text{units}.$

The part of the semicircle included in the shaded region is calculated as:

• Area of semicircle = $\frac{1}{2} \pi r^2$, and
• We subtract the triangular area (formed by the straight line).

Let me calculate the exact shaded area mathematically. One moment! sympy as sp

Given values

radius = 3 # radius of the semicircle rect_length = 6 # length of the rectangle rect_width = 4 # width of the rectangle

Area of the rectangle

rect_area = rect_length * rect_width

Area of the semicircle

semi_circle_area = (1 / 2) * sp.pi * radius**2

Triangle dimensions for subtraction (base = 6, height = 4)

triangle_area = (1 / 2) * rect_length * rect_width

Shaded area = semicircle area - triangle area

shaded_area = semi_circle_area - triangle_area

rect_area, semi_circle_area, triangle_area, shaded_area.simplify()Here are the calculations:

1. Area of the rectangle:
   $\text{Area of rectangle} = 24 \, \text{square units}.$

2. Area of the semicircle:
   $\text{Area of semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (3^2) = 4.5\pi \, \text{square units}.$

3. Area of the triangle:
   The triangle's base is $6$ and height is $4$:
   $\text{Area of triangle} = \frac{1}{2} \times 6 \times 4 = 12 \, \text{square units}.$

4. Shaded area:
   $\text{Shaded area} = \text{Area of semicircle} - \text{Area of triangle} = 4.5\pi - 12 \, \text{square units}.$

Thus, the shaded area is approximately:
$\text{Shaded area} \approx -12 + 4.5\pi \, \text{square units}.$

If you want the numerical value, let me compute it! Numerical value of the shaded area shaded_area_numeric = shaded_area.evalf() shaded_area_numericThe shaded area is approximately:
$2.14 \, \text{square units}.$

Let me know if you'd like further explanation or calculations!

Related Questions:

1. How do you calculate the area of a segment of a circle directly?
2. How does the formula for the area of a triangle apply in this problem?
3. What would happen to the shaded area if the rectangle's dimensions changed?
4. How is the value of $\pi$ approximated in practical problems?
5. How can the concept of integration be applied to find the shaded area?

Tip:

Always sketch and label the geometric shapes in word problems to visualize and simplify the solution process.