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Solve for the perpendicular distance between PQ and SR, given A = 28 cm2, PQ = 8 cm, SR = 6 cm. Find the least mass of tea leaves in the bag for packet sizes 40 g, 250 g, and 350 g.

Mathematics

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Answer

4 cm4 \text{ cm}

7000 grams7000 \text{ grams}

Step-by-step solution

8 steps

here are the solutions for questions 4 and 5:

4.

  1. Write down the formula for the area of a trapezium.

    The area AA of a trapezium with parallel sides aa and bb and perpendicular height hh is given by:

    A=12(a+b)hA = \frac{1}{2}(a+b)h

  2. Identify the given values.

    The parallel sides are PQ=8 cmPQ = 8 \text{ cm} and SR=6 cmSR = 6 \text{ cm}. So, a=8a=8 and b=6b=6.
    The area A=28 cm2A = 28 \text{ cm}^2.
    We need to find the perpendicular distance hh.

  3. Substitute the values into the formula.

    28=12(8+6)h28 = \frac{1}{2}(8+6)h 28=12(14)h28 = \frac{1}{2}(14)h

  4. Simplify the equation.

    28=7h28 = 7h

  5. Solve for hh.

    Divide both sides by 7:

    h=287h = \frac{28}{7} h=4 cmh = 4 \text{ cm}

    The perpendicular distance between PQ and SR is:

    4 cm\boxed{4 \text{ cm}}

    5.
    To determine the least mass of tea leaves in the bag, we need to find the Least Common Multiple (LCM) of the possible packet sizes: 40 g, 250 g, and 350 g. This is because the total mass must be divisible by each of these amounts.

  6. Find the prime factorization of each number.

    40=2×2×2×5=23×5140 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5^1 250=2×5×5×5=21×53250 = 2 \times 5 \times 5 \times 5 = 2^1 \times 5^3 350=2×5×5×7=21×52×71350 = 2 \times 5 \times 5 \times 7 = 2^1 \times 5^2 \times 7^1

  7. Calculate the LCM by taking the highest power of each prime factor present in any of the numbers.

    The prime factors are 2, 5, and 7.
    Highest power of 2: 232^3 (from 40)
    Highest power of 5: 535^3 (from 250)
    Highest power of 7: 717^1 (from 350)

  8. Multiply these highest powers together to find the LCM.

    LCM(40,250,350)=23×53×71\text{LCM}(40, 250, 350) = 2^3 \times 5^3 \times 7^1 =8×125×7= 8 \times 125 \times 7 =1000×7= 1000 \times 7 =7000= 7000

    The least mass of the tea leaves in the bag is:

    7000 grams\boxed{7000 \text{ grams}}
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