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A varies directly as B and inversely as the square of C. Given A = 125 when B = 180 and C = 6. (c) Write down an equation connecting A, B and C. (d) Find the value of C when B = 96 and A = 150. (e) Fi

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3 parts
c

A=25BC2A = \frac{25B}{C^2}

d

C=4C = 4

e

Percentage increase=−16.82%\text{Percentage increase} = -16.82\%

Step-by-step solution

11 steps

here's the solution for question 24:

The relationship between A, B, and C is given by: A varies directly as B and inversely as the square of C.
This can be written as:

A=kBC2A = k \frac{B}{C^2}

where kk is the constant of proportionality.

Part c· 3 steps

Write down an equation connecting A, B and C.

Given A=125A = 125 when B=180B = 180 and C=6C = 6.

  1. Substitute the given values into the variation equation to find kk.

    125=k18062125 = k \frac{180}{6^2} 125=k18036125 = k \frac{180}{36} 125=k×5125 = k \times 5

  2. Solve for kk.

    k=1255k = \frac{125}{5} k=25k = 25

  3. Write the equation connecting A, B, and C using the value of kk.

    A=25BC2\boxed{A = \frac{25B}{C^2}}

Part d

C=4C = 4

Find the value of C when B = 96 and A = 150.

  1. Use the equation found in part c).

    A=25BC2A = \frac{25B}{C^2}

  2. Substitute A=150A = 150 and B=96B = 96 into the equation.

    150=25×96C2150 = \frac{25 \times 96}{C^2}

  3. Solve for C2C^2.

    150C2=25×96150 C^2 = 25 \times 96 150C2=2400150 C^2 = 2400 C2=2400150C^2 = \frac{2400}{150} C2=16C^2 = 16

  4. Solve for CC.

    C=16C = \sqrt{16} C=4\boxed{C = 4}

Part e

Percentage increase=−16.82%\text{Percentage increase} = -16.82\%

Find the percentage increase in A if B increases by 10% and C increases by 15%.

Let the original values be A1A_1, B1B_1, C1C_1.

A1=25B1C12A_1 = \frac{25B_1}{C_1^2}

  1. Calculate the new values of B and C.

    If B increases by 10%, the new B, B2B_2, is B1×(1+0.10)=1.1B1B_1 \times (1 + 0.10) = 1.1 B_1.
    If C increases by 15%, the new C, C2C_2, is C1×(1+0.15)=1.15C1C_1 \times (1 + 0.15) = 1.15 C_1.

  2. Calculate the new value of A, A2A_2.

    A2=25B2C22A_2 = \frac{25B_2}{C_2^2} A2=25(1.1B1)(1.15C1)2A_2 = \frac{25(1.1 B_1)}{(1.15 C_1)^2} A2=25×1.1B11.152C12A_2 = \frac{25 \times 1.1 B_1}{1.15^2 C_1^2} A2=1.11.152×25B1C12A_2 = \frac{1.1}{1.15^2} \times \frac{25 B_1}{C_1^2}

  3. Substitute A1=25B1C12A_1 = \frac{25B_1}{C_1^2} into the equation for A2A_2.

    A2=1.11.152A1A_2 = \frac{1.1}{1.15^2} A_1 A2=1.11.3225A1A_2 = \frac{1.1}{1.3225} A_1 A2≈0.83175879A1A_2 \approx 0.83175879 A_1

  4. Calculate the percentage increase.

    Percentage increase=(A2−A1A1)×100%\text{Percentage increase} = \left(\frac{A_2 - A_1}{A_1}\right) \times 100\% Percentage increase=(A2A1−1)×100%\text{Percentage increase} = \left(\frac{A_2}{A_1} - 1\right) \times 100\% Percentage increase=(1.11.3225−1)×100%\text{Percentage increase} = \left(\frac{1.1}{1.3225} - 1\right) \times 100\% Percentage increase=(0.83175879−1)×100%\text{Percentage increase} = (0.83175879 - 1) \times 100\% Percentage increase=−0.16824121×100%\text{Percentage increase} = -0.16824121 \times 100\% Percentage increase≈−16.82%\text{Percentage increase} \approx -16.82\%

    The negative sign indicates a decrease.

    Percentage increase=−16.82%\boxed{\text{Percentage increase} = -16.82\%}
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