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A regular hexagonal flower bed of side 3 m is to be paved. The bed can be divided into six identical triangles, each with a base of 1 m and a perpendicular height of 2.6 m. (a) Calculate the area of one triangle.

Mathematics

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Question 31:

a) Calculate the import duty.

  1. Identify the customs value and the import duty percentage.

    Customs value = KSh 80,000
    Import duty = 25% of the customs value

  2. Calculate the import duty.

    Import Duty=0.25×KSh 80,000\text{Import Duty} = 0.25 \times \text{KSh } 80,000 Import Duty=KSh 20,000\text{Import Duty} = \text{KSh } 20,000

    The import duty is KSh 20,000.

    b) Calculate the VAT.

  3. Calculate the sum of the customs value and the import duty.

    Sum=Customs Value+Import Duty\text{Sum} = \text{Customs Value} + \text{Import Duty} Sum=KSh 80,000+KSh 20,000\text{Sum} = \text{KSh } 80,000 + \text{KSh } 20,000 Sum=KSh 100,000\text{Sum} = \text{KSh } 100,000

  4. Calculate the VAT, which is 16% of this sum.

    VAT=0.16×KSh 100,000\text{VAT} = 0.16 \times \text{KSh } 100,000 VAT=KSh 16,000\text{VAT} = \text{KSh } 16,000

    The VAT is KSh 16,000.

    c) Find the total amount payable for the machine, including duty and VAT.

  5. Add the customs value, import duty, and VAT.

    Total Amount Payable=Customs Value+Import Duty+VAT\text{Total Amount Payable} = \text{Customs Value} + \text{Import Duty} + \text{VAT} Total Amount Payable=KSh 80,000+KSh 20,000+KSh 16,000\text{Total Amount Payable} = \text{KSh } 80,000 + \text{KSh } 20,000 + \text{KSh } 16,000 Total Amount Payable=KSh 116,000\text{Total Amount Payable} = \text{KSh } 116,000

    The total amount payable is KSh 116,000\boxed{\text{KSh } 116,000}.

    Question 32:

    a) Factorize 12ab−8a12ab - 8a.

  6. Find the greatest common factor (GCF) of $12ab$ and $8a$.

    The GCF of 12 and 8 is 4.
    The common variable factor is aa.
    So, the GCF is 4a4a.

  7. Factor out the GCF from the expression.

    12ab−8a=4a(3b)−4a(2)12ab - 8a = 4a(3b) - 4a(2) 12ab−8a=4a(3b−2)12ab - 8a = 4a(3b - 2)

    The factored expression is 4a(3b−2)\boxed{4a(3b - 2)}.

    b) Simplify 2x3+x4\frac{2x}{3} + \frac{x}{4}.

  8. Find a common denominator for the fractions.

    The least common multiple of 3 and 4 is 12.

  9. Rewrite each fraction with the common denominator.

    2x3=2x×43×4=8x12\frac{2x}{3} = \frac{2x \times 4}{3 \times 4} = \frac{8x}{12} x4=x×34×3=3x12\frac{x}{4} = \frac{x \times 3}{4 \times 3} = \frac{3x}{12}

  10. Add the fractions.

    8x12+3x12=8x+3x12\frac{8x}{12} + \frac{3x}{12} = \frac{8x + 3x}{12} 8x+3x12=11x12\frac{8x + 3x}{12} = \frac{11x}{12}

    The simplified expression is 11x12\boxed{\frac{11x}{12}}.

    c) Evaluate 3a2−2b3a^2 - 2b when a=−2a = -2 and b=5b = 5.

  11. Substitute the given values of aa and bb into the expression.

    3a2−2b=3(−2)2−2(5)3a^2 - 2b = 3(-2)^2 - 2(5)

  12. Perform the exponentiation.

    3(−2)2−2(5)=3(4)−2(5)3(-2)^2 - 2(5) = 3(4) - 2(5)

  13. Perform the multiplications.

    3(4)−2(5)=12−103(4) - 2(5) = 12 - 10

  14. Perform the subtraction.

    12−10=212 - 10 = 2

    The evaluated value is 2\boxed{2}.

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