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Identify the correct formula for the area of a triangle.

Mathematics

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Answer

15.16 cm215.16 \text{ cm}^2

Step-by-step solution

5 steps

The student's calculation uses an incorrect formula for the area of a shaded region. The formula 12×(5+7)×cos⁡60∘\frac{1}{2} \times (5+7) \times \cos 60^\circ is not a standard area formula, and the unit cm3\text{cm}^3 is for volume, not area.

Assuming the shaded region is a triangle with two sides measuring 5 cm5 \text{ cm} and 7 cm7 \text{ cm} and an included angle of 60∘60^\circ.

  1. Identify the correct formula for the area of a triangle.

    The area of a triangle given two sides aa and bb and the included angle CC is:

    Area=12absin⁡C\text{Area} = \frac{1}{2}ab \sin C

  2. Substitute the given values into the formula.

    Area=12×5 cm×7 cm×sin⁡60∘\text{Area} = \frac{1}{2} \times 5 \text{ cm} \times 7 \text{ cm} \times \sin 60^\circ

  3. Calculate the value of sin⁡60∘\sin 60^\circ.

    sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2}

  4. Perform the multiplication.

    Area=12×35 cm2×32\text{Area} = \frac{1}{2} \times 35 \text{ cm}^2 \times \frac{\sqrt{3}}{2} Area=3534 cm2\text{Area} = \frac{35\sqrt{3}}{4} \text{ cm}^2

  5. Calculate the numerical value and round to two decimal places.

    Area≈35×1.732054 cm2\text{Area} \approx \frac{35 \times 1.73205}{4} \text{ cm}^2 Area≈60.621754 cm2\text{Area} \approx \frac{60.62175}{4} \text{ cm}^2 Area≈15.1554 cm2\text{Area} \approx 15.1554 \text{ cm}^2 Area≈15.16 cm2\text{Area} \approx \boxed{15.16 \text{ cm}^2}
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