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Question 2: The volume of a sphere is 38808 cm3. Find the radius. Use pi = 22/7.

Mathematics

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Answers

2 parts
2

21 cm21 \text{ cm}

3

6415y85k145z6564^{\frac{1}{5}} y^{\frac{8}{5}} k^{\frac{14}{5}} z^{\frac{6}{5}}

Step-by-step solution

11 steps

I can see two questions in the image. I will solve question 2 and question 3.

Part 2· 5 steps

The volume of a sphere

  1. Write down the given formula and values.

    The volume of a sphere is given by V=43πr3V = \frac{4}{3}\pi r^3.
    Given volume V=38808 cm3V = 38808 \text{ cm}^3.
    We need to find the radius rr.
    We will use π=227\pi = \frac{22}{7}.

  2. Substitute the given values into the formula.

    38808=43×227×r338808 = \frac{4}{3} \times \frac{22}{7} \times r^3
  3. Isolate r3r^3.

    38808=8821r338808 = \frac{88}{21} r^3

    Multiply both sides by 2188\frac{21}{88}:

    r3=38808×2188r^3 = 38808 \times \frac{21}{88}
  4. Perform the multiplication and division.

    First, divide 38808 by 88:

    38808÷88=44138808 \div 88 = 441

    Now, multiply by 21:

    r3=441×21r^3 = 441 \times 21 r3=9261r^3 = 9261
  5. Find the cube root of 9261 to get rr.

    r=92613r = \sqrt[3]{9261}

    To find the cube root, we can try small integer values or factorize.
    We know 203=800020^3 = 8000 and 213=926121^3 = 9261.

    r=21 cmr = 21 \text{ cm}

    The radius of the sphere is 21 cm\boxed{21 \text{ cm}}.


Part 3

6415y85k145z6564^{\frac{1}{5}} y^{\frac{8}{5}} k^{\frac{14}{5}} z^{\frac{6}{5}}

Solve the expression

  1. Rewrite the expression using fractional exponents.

    The expression is (32y10k15)2z1016y12k16z45\sqrt[5]{\frac{(32y^{10}k^{15})^2 z^{10}}{16y^{12}k^{16}z^4}}.
    This can be written as ((32y10k15)2z1016y12k16z4)15\left(\frac{(32y^{10}k^{15})^2 z^{10}}{16y^{12}k^{16}z^4}\right)^{\frac{1}{5}}.

  2. Simplify the numerator inside the parenthesis.

    (32y10k15)2=322(y10)2(k15)2(32y^{10}k^{15})^2 = 32^2 (y^{10})^2 (k^{15})^2 =1024y10×2k15×2= 1024 y^{10 \times 2} k^{15 \times 2} =1024y20k30= 1024 y^{20} k^{30}

    So the numerator becomes 1024y20k30z101024 y^{20} k^{30} z^{10}.

  3. Substitute the simplified numerator back into the expression.

    (1024y20k30z1016y12k16z4)15\left(\frac{1024 y^{20} k^{30} z^{10}}{16y^{12}k^{16}z^4}\right)^{\frac{1}{5}}
  4. Simplify the terms inside the parenthesis by dividing coefficients and subtracting exponents for like bases.

    For the coefficients: 102416=64\frac{1024}{16} = 64.
    For yy: y20y12=y20−12=y8\frac{y^{20}}{y^{12}} = y^{20-12} = y^8.
    For kk: k30k16=k30−16=k14\frac{k^{30}}{k^{16}} = k^{30-16} = k^{14}.
    For zz: z10z4=z10−4=z6\frac{z^{10}}{z^4} = z^{10-4} = z^6.

    So the expression inside the parenthesis simplifies to 64y8k14z664 y^8 k^{14} z^6.

  5. Apply the 15\frac{1}{5} exponent to each term.

    (64y8k14z6)15(64 y^8 k^{14} z^6)^{\frac{1}{5}} =6415(y8)15(k14)15(z6)15= 64^{\frac{1}{5}} (y^8)^{\frac{1}{5}} (k^{14})^{\frac{1}{5}} (z^6)^{\frac{1}{5}} =6415y85k145z65= 64^{\frac{1}{5}} y^{\frac{8}{5}} k^{\frac{14}{5}} z^{\frac{6}{5}}
  6. Calculate 641564^{\frac{1}{5}}.

    Since 25=322^5 = 32 and 35=2433^5 = 243, 641564^{\frac{1}{5}} is not an integer.
    We can leave it in this form or approximate it. For exact solution, we leave it as 641564^{\frac{1}{5}}.

    The simplified expression is 6415y85k145z65\boxed{64^{\frac{1}{5}} y^{\frac{8}{5}} k^{\frac{14}{5}} z^{\frac{6}{5}}}.

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