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c) The diagram shows a circle centre O. If POQ = x°, the diameter of the circle is 7cm and the area of the shaded portion is 27.5cm². Find, correct to the nearest degree, the value of x. [Take π = 22/7]

Mathematics

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Answers

3 parts
c

103∘103^\circ

a

30.43%30.43\%

b
250

Step-by-step solution

12 steps

Part c· 4 steps

Step 1: Find the radius of the circle.

The diameter is 7 cm7 \text{ cm}, so the radius r=72 cmr = \frac{7}{2} \text{ cm}.

  1. Calculate the area of the entire circle.

    The area of a circle is given by A=πr2A = \pi r^2.

    A=227×(72)2A = \frac{22}{7} \times \left(\frac{7}{2}\right)^2 A=227×494A = \frac{22}{7} \times \frac{49}{4} A=22×74=1544=38.5 cm2A = \frac{22 \times 7}{4} = \frac{154}{4} = 38.5 \text{ cm}^2
  2. Determine the area of the unshaded sector.

    The area of the shaded portion is 27.5 cm227.5 \text{ cm}^2.
    Area of unshaded sector = Total area - Area of shaded portion

    Area of unshaded sector=38.5−27.5=11 cm2\text{Area of unshaded sector} = 38.5 - 27.5 = 11 \text{ cm}^2
  3. Use the area of the unshaded sector to find the angle xx.

    The area of a sector with angle θ\theta is given by θ360∘×πr2\frac{\theta}{360^\circ} \times \pi r^2.
    Here, the angle of the unshaded sector is x∘x^\circ.

    11=x360×38.511 = \frac{x}{360} \times 38.5 x=11×36038.5x = \frac{11 \times 360}{38.5} x=396038.5x = \frac{3960}{38.5} x=102.857...x = 102.857...
  4. Round the value of xx to the nearest degree.

    x≈103∘x \approx 103^\circ

    The value of xx is 103∘\boxed{103^\circ}.

Part a

30.43%30.43\%

Step 1: Calculate the import duty.

Original cost of fish = GHe 400.00.
Import duty = 15% of GHe 400.00.

Import Duty=0.15×400=GHe 60.00\text{Import Duty} = 0.15 \times 400 = \text{GHe } 60.00
  1. Calculate the cost after import duty.

    Cost after duty=400+60=GHe 460.00\text{Cost after duty} = 400 + 60 = \text{GHe } 460.00
  2. Calculate the sales tax.

    Sales tax = 10% of the total cost including import duty.

    Sales Tax=0.10×460=GHe 46.00\text{Sales Tax} = 0.10 \times 460 = \text{GHe } 46.00
  3. Calculate the total cost price (CP).

    CP=Cost after duty+Sales Tax\text{CP} = \text{Cost after duty} + \text{Sales Tax} CP=460+46=GHe 506.00\text{CP} = 460 + 46 = \text{GHe } 506.00
  4. Calculate the profit.

    Selling Price (SP) = GHe 660.00.

    Profit=SP−CP\text{Profit} = \text{SP} - \text{CP} Profit=660−506=GHe 154.00\text{Profit} = 660 - 506 = \text{GHe } 154.00
  5. Calculate the percentage profit.

    Percentage Profit=ProfitCP×100%\text{Percentage Profit} = \frac{\text{Profit}}{\text{CP}} \times 100\% Percentage Profit=154506×100%\text{Percentage Profit} = \frac{154}{506} \times 100\% Percentage Profit=0.304347...×100%\text{Percentage Profit} = 0.304347... \times 100\% Percentage Profit≈30.43%\text{Percentage Profit} \approx 30.43\%

    The percentage profit is 30.43%\boxed{30.43\%}.

Part b250

Step 1: Define variables and total students.

Number of boys = 1000.
Let the number of girls be GG.
Total number of students = 1000+G1000 + G.

  1. Calculate the number of successful students from boys and girls.

    Number of successful boys = 50% of 1000 = 0.50×1000=5000.50 \times 1000 = 500.
    Number of successful girls = 40% of G=0.40GG = 0.40G.
    Total successful students = 500+0.40G500 + 0.40G.

  2. Calculate the total number of successful students using the overall percentage.

    48% of the total number of students were successful.
    Total successful students = 0.48×(1000+G)0.48 \times (1000 + G).

  3. Set up and solve the equation for GG.

    500+0.40G=0.48(1000+G)500 + 0.40G = 0.48(1000 + G) 500+0.40G=480+0.48G500 + 0.40G = 480 + 0.48G

    Subtract 0.40G0.40G from both sides:

    500=480+0.48G−0.40G500 = 480 + 0.48G - 0.40G 500=480+0.08G500 = 480 + 0.08G

    Subtract 480 from both sides:

    500−480=0.08G500 - 480 = 0.08G 20=0.08G20 = 0.08G

    Divide by 0.08:

    G=200.08G = \frac{20}{0.08} G=250G = 250

    The number of girls in the school is 250\boxed{250}.

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