Calculate the area of the outer rectangle.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Okay, Nmk, let's calculate the shaded areas for each figure. We'll use $\pi = 3.1$.
a)
Step 1: Calculate the area of the outer rectangle.
The length is $10$ cm and the width is $6$ cm.
$$ \text{Area}_{\text{outer}} = 10 \text{ cm} \times 6 \text{ cm} = 60 \text{ cm}^2 $$
Step 2: Calculate the area of the inner (unshaded) rectangle.
The length is $6$ cm and the width is $4$ cm.
$$ \text{Area}_{\text{inner}} = 6 \text{ cm} \times 4 \text{ cm} = 24 \text{ cm}^2 $$
Step 3: Calculate the shaded area.
The shaded area is the area of the outer rectangle minus the area of the inner rectangle.
$$ \text{Area}_{\text{shaded}} = \text{Area}_{\text{outer}} - \text{Area}_{\text{inner}} = 60 \text{ cm}^2 - 24 \text{ cm}^2 = \boxed{36 \text{ cm}^2} $$
b)
Step 1: Calculate the area of the rectangular part.
The length is $12$ cm and the width is $12$ cm.
$$ \text{Area}_{\text{rectangle}} = 12 \text{ cm} \times 12 \text{ cm} = 144 \text{ cm}^2 $$
Step 2: Calculate the area of the semicircular part.
The diameter of the semicircle is $12$ cm, so the radius is $r = \frac{12}{2} = 6$ cm.
$$ \text{Area}_{\text{semicircle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \times 3.1 \times (6 \text{ cm})^2 $$
$$ \text{Area}_{\text{semicircle}} = \frac{1}{2} \times 3.1 \times 36 \text{ cm}^2 = 3.1 \times 18 \text{ cm}^2 = 55.8 \text{ cm}^2 $$
Step 3: Calculate the total shaded area.
The total shaded area is the sum of the rectangle's area and the semicircle's area.
$$ \text{Area}_{\text{total shaded}} = \text{Area}_{\text{rectangle}} + \text{Area}_{\text{semicircle}} = 144 \text{ cm}^2 + 55.8 \text{ cm}^2 = \boxed{199.8 \text{ cm}^2} $$
c)
Step 1: Calculate the area of the square.
The side length of the square is $4 \text{ cm} + 4 \text{ cm} = 8$ cm.
$$ \text{Area}_{\text{square}} = (8 \text{ cm})^2 = 64 \text{ cm}^2 $$
Step 2: Calculate the area of the unshaded quarter circle.
The radius of the quarter circle is $4$ cm.
$$ \text{Area}_{\text{quarter circle}} = \frac{1}{4} \pi r^2 = \frac{1}{4} \times 3.1 \times (4 \text{ cm})^2 $$
$$ \text{Area}_{\text{quarter circle}} = \frac{1}{4} \times 3.1 \times 16 \text{ cm}^2 = 3.1 \times 4 \text{ cm}^2 = 12.4 \text{ cm}^2 $$
Step 3: Calculate the shaded area.
The shaded area is the area of the square minus the area of the quarter circle.
$$ \text{Area}_{\text{shaded}} = \text{Area}_{\text{square}} - \text{Area}_{\text{quarter circle}} = 64 \text{ cm}^2 - 12.4 \text{ cm}^2 = \boxed{51.6 \text{ cm}^2} $$
d)
Step 1: Calculate the area of the outer circle.
The diameter of the outer circle is $20$ cm, so the radius is $R = \frac{20}{2} = 10$ cm.
$$ \text{Area}_{\text{outer}} = \pi R^2 = 3.1 \times (10 \text{ cm})^2 $$
$$ \text{Area}_{\text{outer}} = 3.1 \times 100 \text{ cm}^2 = 310 \text{ cm}^2 $$
Step 2: Calculate the area of the inner (unshaded) circle.
The diameter of the inner circle is $10$ cm, so the radius is $r = \frac{10}{2} = 5$ cm.
$$ \text{Area}_{\text{inner}} = \pi r^2 = 3.1 \times (5 \text{ cm})^2 $$
$$ \text{Area}_{\text{inner}} = 3.1 \times 25 \text{ cm}^2 = 77.5 \text{ cm}^2 $$
Step 3: Calculate the shaded area.
The shaded area is the area of the outer circle minus the area of the inner circle.
$$ \text{Area}_{\text{shaded}} = \text{Area}_{\text{outer}} - \text{Area}_{\text{inner}} = 310 \text{ cm}^2 - 77.5 \text{ cm}^2 = \boxed{232.5 \text{ cm}^2} $$
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