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Given A = k(B2/C2), solve for A and percentage change in A under specified conditions.

Mathematics

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2 parts
i
5.08
ii

251.56%251.56\%

Step-by-step solution

5 steps

Here's the solution to question 3, parts i) and ii):

From the previous question, the relationship between A, B, and C is:

A=kB2C2A = k \frac{B^2}{C^2}

Part i· 2 steps

Given that when A=20, B=5 and C=9. Find A when B=7 and C=25.

  1. Find the constant kk using the initial values A=20,B=5,C=9A=20, B=5, C=9.

    Substitute these values into the equation:

    20=k529220 = k \frac{5^2}{9^2} 20=k258120 = k \frac{25}{81}

    To solve for kk, multiply both sides by 8125\frac{81}{25}:

    k=20×8125k = 20 \times \frac{81}{25} k=4×5×815×5k = \frac{4 \times 5 \times 81}{5 \times 5} k=4×815k = \frac{4 \times 81}{5} k=3245k = \frac{324}{5} k=64.8k = 64.8
  2. Use the value of kk to find AA when B=7B=7 and C=25C=25.

    Substitute k=64.8k=64.8, B=7B=7, and C=25C=25 into the equation:

    A=64.872252A = 64.8 \frac{7^2}{25^2} A=64.849625A = 64.8 \frac{49}{625} A=64.8×49625A = \frac{64.8 \times 49}{625} A=3175.2625A = \frac{3175.2}{625} A=5.08032A = 5.08032

    Rounding to a reasonable number of decimal places, for example, two:

    A≈5.08A \approx 5.08

    The value of A when B=7 and C=25 is 5.08\boxed{5.08}.

Part ii

251.56%251.56\%

If B is increased by 20% and C decreased by 36%, find the percentage change in A.

  1. Express the new values of B and C in terms of their original values.

    Let the original values be B1B_1 and C1C_1.
    If B is increased by 20%, the new value B2B_2 is:

    B2=B1+0.20B1=1.20B1B_2 = B_1 + 0.20 B_1 = 1.20 B_1

    If C is decreased by 36%, the new value C2C_2 is:

    C2=C1−0.36C1=0.64C1C_2 = C_1 - 0.36 C_1 = 0.64 C_1
  2. Write the original and new expressions for A.

    Original A (A1A_1):

    A1=kB12C12A_1 = k \frac{B_1^2}{C_1^2}

    New A (A2A_2):

    A2=kB22C22A_2 = k \frac{B_2^2}{C_2^2}

    Substitute B2=1.20B1B_2 = 1.20 B_1 and C2=0.64C1C_2 = 0.64 C_1 into the equation for A2A_2:

    A2=k(1.20B1)2(0.64C1)2A_2 = k \frac{(1.20 B_1)^2}{(0.64 C_1)^2} A2=k1.44B120.4096C12A_2 = k \frac{1.44 B_1^2}{0.4096 C_1^2} A2=1.440.4096(kB12C12)A_2 = \frac{1.44}{0.4096} \left( k \frac{B_1^2}{C_1^2} \right)

    Recognize that kB12C12k \frac{B_1^2}{C_1^2} is A1A_1:

    A2=1.440.4096A1A_2 = \frac{1.44}{0.4096} A_1 A2=3.515625A1A_2 = 3.515625 A_1
  3. Calculate the percentage change in A.

    Percentage change = New Value−Original ValueOriginal Value×100%\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%

    Percentage change in A=A2−A1A1×100%\text{Percentage change in A} = \frac{A_2 - A_1}{A_1} \times 100\% Percentage change in A=3.515625A1−A1A1×100%\text{Percentage change in A} = \frac{3.515625 A_1 - A_1}{A_1} \times 100\% Percentage change in A=(3.515625−1)×100%\text{Percentage change in A} = (3.515625 - 1) \times 100\% Percentage change in A=2.515625×100%\text{Percentage change in A} = 2.515625 \times 100\% Percentage change in A=251.5625%\text{Percentage change in A} = 251.5625\%

    Rounding to two decimal places:

    Percentage change in A≈251.56%\text{Percentage change in A} \approx 251.56\%

    The percentage change in A is an increase of 251.56%\boxed{251.56\%}.

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