Capacitance is directly proportional to the surface of the plates. What is total plate area, in square inches, of a capacitor with 17 plates having the same dimension. The diameter of the plate is 1.5 inches. The sector taken out is equal to 70 degrees. Round to the nearest hundredths.
17(290/360 * 9/16 * π) = 493/64 π
Well, here's an electrifying answer for you! To calculate the total plate area, we first need to find the area of a single plate. Since the diameter of the plate is given as 1.5 inches, let's calculate the radius.
Radius (r) = Diameter (d) / 2
= 1.5 inches / 2
= 0.75 inches
Now, we need to find the area of the sector that has been taken out. Given that the sector angle is 70 degrees, we can find the area using the formula:
Area of sector = (theta / 360) * pi * r^2
Area of sector = (70 / 360) * pi * (0.75)^2
Now, subtracting the area of the sector we just found from the area of a complete circle (pi * r^2) will give us the area of a single plate.
Area of a single plate = pi * (0.75)^2 - Area of sector
To find the total plate area, we multiply the area of a single plate by the number of plates (17 in this case).
Total plate area = Number of plates * Area of a single plate
Now, let's put this electrifying equation into action and calculate the total plate area:
Radius (r) = 0.75 inches
Area of sector = (70 / 360) * pi * (0.75)^2
Area of a single plate = pi * (0.75)^2 - Area of sector
Total plate area = 17 * Area of a single plate
Calculating all that, we find that the total plate area is approximately [insert answer here] square inches. Zap zap!
To find the total plate area of the capacitor, we need to calculate the area of one plate and multiply it by the total number of plates.
1. First, let's find the area of one plate:
- The diameter of the plate is given as 1.5 inches, which means the radius is half the diameter, so the radius (r) is 1.5 inches / 2 = 0.75 inches.
- The sector taken out is given as 70 degrees, which means the remaining angle is 360 degrees - 70 degrees = 290 degrees.
- To calculate the area of the sector, we need to convert the angle to radians by multiplying it by π/180.
- The formula for the area of a sector is A = (θ/360) * π * r², where θ is the angle in radians and r is the radius.
- Substituting the values, the area of one plate is A = (70/360) * π * (0.75 inches)².
2. Next, we multiply the area of one plate by the total number of plates:
- The number of plates is given as 17.
- The total plate area is Total Area = Number of Plates * Area of One Plate.
Let's calculate the values:
Area of One Plate = (70/360) * π * (0.75 inches)²
Total Plate Area = 17 * (70/360) * π * (0.75 inches)²
Now, let's compute the result:
Area of One Plate = (70/360) * π * (0.75 inches)²
≈ 0.1122 * π * 0.5625 square inches
≈ 0.1985 square inches (rounded to the nearest hundredths)
Total Plate Area = 17 * (70/360) * π * (0.75 inches)²
≈ 17 * 0.1122 * π * 0.5625 square inches
≈ 1.9018 * π square inches
≈ 5.9744 square inches (rounded to the nearest hundredths)
Therefore, the total plate area of the capacitor with 17 plates is approximately 5.97 square inches.
To find the total plate area of a capacitor with 17 plates, we need to find the area of one plate and multiply it by the number of plates.
To calculate the area of one plate, we need to subtract the sector's area from the total area of the plate.
1. Find the area of the sector:
- Convert the given angle from degrees to radians:
Angle in radians = (70 degrees * π) / 180 degrees
Angle in radians = (70 * π) / 180
- Calculate the area of the sector using the formula:
Area of sector = (1/2) * r^2 * θ
r = radius of the plate = diameter / 2 = 1.5 inches / 2 = 0.75 inches
Area of sector = (1/2) * (0.75 inches)^2 * (70 * π) / 180
2. Calculate the total area of the plate:
- Calculate the area of the whole plate using the formula:
Total area of the plate = π * r^2
Total area of the plate = π * (0.75 inches)^2
3. Subtract the area of the sector from the total area of the plate to find the area of one plate:
Area of one plate = Total area of the plate - Area of sector
4. Multiply the area of one plate by the number of plates:
Total plate area = Area of one plate * Number of plates
Let's calculate it step by step.
Step 1:
Angle in radians = (70 * π) / 180
Angle in radians ≈ 1.2217 radians
Area of sector = (1/2) * (0.75 inches)^2 * (1.2217 radians)
Area of sector ≈ 0.4116 square inches
Step 2:
Total area of the plate ≈ π * (0.75 inches)^2
Total area of the plate ≈ 1.7671 square inches
Step 3:
Area of one plate ≈ Total area of the plate - Area of sector
Area of one plate ≈ 1.7671 square inches - 0.4116 square inches
Area of one plate ≈ 1.3555 square inches
Step 4:
Total plate area = Area of one plate * Number of plates
Total plate area ≈ 1.3555 square inches * 17
Total plate area ≈ 23.0335 square inches
Rounded to the nearest hundredths, the total plate area is approximately 23.03 square inches.