To calculate the area of the major sector of the circle from which the cone is formed, we first need to find the slant height of the cone using the given curved surface area and radius.
Step 1: Calculate the Slant Height
The formula for the curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi r l \]
Where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.
Given:
- CSA = 440 cm²,
- \( r = 7 \) cm.
Now, we can rearrange the formula to solve for the slant height \( l \):
\[ l = \frac{\text{CSA}}{\pi r} \]
Substituting the values we have:
\[ l = \frac{440}{\pi \times 7} \]
Calculating \( l \):
\[ l = \frac{440}{7\pi} \approx \frac{440}{21.99} \approx 20.00 , \text{cm} \quad (\text{approximating } \pi \approx 3.14) \]
Step 2: Calculate the Length of the Arc of the Sector
The slant height \( l \) of the cone represents the radius of the sector from which the cone is formed. The circumference of the circle corresponding to the sector can be calculated with:
\[ C = 2\pi r = 2\pi \times 7 = 14\pi , \text{cm} \]
When the sector is formed into a cone, the length of the arc of the sector is equal to the circumference of the base of the cone:
\[ \text{Circumference of base of cone} = C_{\text{cone}} = 2\pi \times 7 = 14\pi , \text{cm} \]
Step 3: Find the angle of the sector
Let's denote the angle of the sector as \( \theta \) (in radians). The arc length (which equals the base circumference of the cone) of the sector is given by:
\[ \text{Arc Length} = l \theta = 14\pi \]
Substituting \( l = 20 \) cm into the equation gives:
\[ 20\theta = 14\pi \] \[ \theta = \frac{14\pi}{20} = \frac{7\pi}{10} , \text{radians} \]
Step 4: Area of the Sector
The area \( A \) of a sector of a circle is given by the formula:
\[ A = \frac{1}{2} r^2 \theta \]
Using the radius \( l = 20 \) cm and the angle \( \theta = \frac{7\pi}{10} \):
\[ A = \frac{1}{2} \times 20^2 \times \frac{7\pi}{10} \]
Calculating it step by step:
\[ A = \frac{1}{2} \times 400 \times \frac{7\pi}{10} = 200 \times \frac{7\pi}{10} = 140\pi \]
Step 5: Calculate the Area of the Major Sector
Now to find the area of the major sector, we need the total area of the circle and subtract the area of the minor sector.
The total area of the larger circle (with radius 20 cm) is:
\[ \text{Total Area} = \pi l^2 = \pi \times 20^2 = 400\pi \]
The area of the major sector is given by the total area minus the area of the minor sector:
\[ \text{Area of the Major Sector} = 400\pi - 140\pi = 260\pi \]
Final Calculation
Using \( \pi \approx 3.14 \):
\[ \text{Area of the Major Sector} \approx 260 \times 3.14 \approx 815.6 , \text{cm}^2 \]
Thus, the area of the major sector of the circle from which the cone is formed is approximately 815.6 cm².
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