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Solve for two numbers given their sum and difference, and find the area of a sector.

Mathematics

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Answer

b

x=10,y=8x=10, y=8

37.5π cm2 or approximately 117.81 cm237.5\pi \text{ cm}^2 \text{ or approximately } 117.81 \text{ cm}^2

192 cm2192 \text{ cm}^2

Step-by-step solution

13 steps

Part a

Let the two numbers be xx and yy.

The sum of the two numbers is 18:

x+y=18(1)x + y = 18 \quad (1)

Their difference is 2:

x−y=2(2)x - y = 2 \quad (2)

The set of simultaneous equations is:

{x+y=18x−y=2\begin{cases} x + y = 18 \\ x - y = 2 \end{cases}

Part b

x=10,y=8x=10, y=8

Step 1: Add equation (1) and equation (2).

(x+y)+(x−y)=18+2(x + y) + (x - y) = 18 + 2 2x=202x = 20

  1. Solve for xx.

    x=202x = \frac{20}{2} x=10x = 10

  2. Substitute the value of xx into equation (1).

    10+y=1810 + y = 18

  3. Solve for yy.

    y=18−10y = 18 - 10 y=8y = 8

    The two numbers are 10 and 8.

    x=10,y=8\boxed{x=10, y=8}

  4. Identify the given values for the sector.

    Radius r=15 cmr = 15 \text{ cm}
    Angle θ=60∘\theta = 60^\circ

  5. Use the formula for the area of a sector.

    A=θ360∘πr2A = \frac{\theta}{360^\circ} \pi r^2

  6. Substitute the values into the formula.

    A=60∘360∘π(15 cm)2A = \frac{60^\circ}{360^\circ} \pi (15 \text{ cm})^2 A=16π(225 cm2)A = \frac{1}{6} \pi (225 \text{ cm}^2)

  7. Calculate the area.

    A=2256π cm2A = \frac{225}{6} \pi \text{ cm}^2 A=752π cm2A = \frac{75}{2} \pi \text{ cm}^2 A=37.5π cm2A = 37.5 \pi \text{ cm}^2

    Using π≈3.14159\pi \approx 3.14159:

    A≈37.5×3.14159 cm2A \approx 37.5 \times 3.14159 \text{ cm}^2 A≈117.81 cm2A \approx 117.81 \text{ cm}^2

    The area of the sector is 37.5π cm2 or approximately 117.81 cm2\boxed{37.5\pi \text{ cm}^2 \text{ or approximately } 117.81 \text{ cm}^2}.

  8. Identify the dimensions of the bar soap (cuboid).

    Length L=12 cmL = 12 \text{ cm}
    Width W=4 cmW = 4 \text{ cm}
    Height H=3 cmH = 3 \text{ cm}

  9. Use the formula for the surface area of a cuboid.

    SA=2(LW+LH+WH)SA = 2(LW + LH + WH)

  10. Substitute the dimensions into the formula.

    SA=2((12 cm)(4 cm)+(12 cm)(3 cm)+(4 cm)(3 cm))SA = 2((12 \text{ cm})(4 \text{ cm}) + (12 \text{ cm})(3 \text{ cm}) + (4 \text{ cm})(3 \text{ cm}))

  11. Calculate the products inside the parentheses.

    SA=2(48 cm2+36 cm2+12 cm2)SA = 2(48 \text{ cm}^2 + 36 \text{ cm}^2 + 12 \text{ cm}^2)

  12. Sum the areas inside the parentheses.

    SA=2(96 cm2)SA = 2(96 \text{ cm}^2)

  13. Calculate the total surface area.

    SA=192 cm2SA = 192 \text{ cm}^2

    The surface area of the bar soap is 192 cm2\boxed{192 \text{ cm}^2}.

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