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The image shows a diagram with dimensions labeled as 45y, 20y, 15x, 30y, 10x, 50x, and 4. The text "Perimeters." is also visible. The question is to find the perimeters.

Mathematics

ScanSolve AI Answer

Answer

30+42+21330 + 4\sqrt{2} + 2\sqrt{13}

Step-by-step solution

7 steps

  1. Identify the given lengths.

    The lengths of the parallel bases are 10 m (top) and 20 m (bottom).
    The height is h=4h = 4 m.
    One base angle is θ=45∘\theta = 45^\circ (between the longer base and one leg).

  2. Find the difference in base lengths.

    Difference=20 m−10 m=10 m\text{Difference} = 20\,\text{m} - 10\,\text{m} = 10\,\text{m}

    This difference equals the sum of the horizontal projections of the two legs: p1+p2=10p_1 + p_2 = 10 m.

  3. Calculate the left projection p1p_1 (for the 45∘45^\circ angle).

    Formula: tan⁡θ=hp1\tan \theta = \frac{h}{p_1}

    p1=htan⁡45∘=4 m1=4 mp_1 = \frac{h}{\tan 45^\circ} = \frac{4\,\text{m}}{1} = 4\,\text{m}

  4. Calculate the right projection p2p_2.

    p2=10 m−4 m=6 mp_2 = 10\,\text{m} - 4\,\text{m} = 6\,\text{m}

  5. Calculate the left leg length l1l_1.

    Formula: l1=h2+p12l_1 = \sqrt{h^2 + p_1^2}

    l1=(4 m)2+(4 m)2=16 m2+16 m2=32 m2=16⋅2 m=42 ml_1 = \sqrt{(4\,\text{m})^2 + (4\,\text{m})^2} = \sqrt{16\,\text{m}^2 + 16\,\text{m}^2} = \sqrt{32\,\text{m}^2} = \sqrt{16 \cdot 2}\,\text{m} = 4\sqrt{2}\,\text{m}

  6. Calculate the right leg length l2l_2.

    Formula: l2=h2+p22l_2 = \sqrt{h^2 + p_2^2}

    l2=(4 m)2+(6 m)2=16 m2+36 m2=52 m2=4⋅13 m=213 ml_2 = \sqrt{(4\,\text{m})^2 + (6\,\text{m})^2} = \sqrt{16\,\text{m}^2 + 36\,\text{m}^2} = \sqrt{52\,\text{m}^2} = \sqrt{4 \cdot 13}\,\text{m} = 2\sqrt{13}\,\text{m}

  7. Calculate the perimeter PP.

    Formula: P=10 m+20 m+l1+l2P = 10\,\text{m} + 20\,\text{m} + l_1 + l_2

    P=10 m+20 m+42 m+213 m=30+42+213 mP = 10\,\text{m} + 20\,\text{m} + 4\sqrt{2}\,\text{m} + 2\sqrt{13}\,\text{m} = 30 + 4\sqrt{2} + 2\sqrt{13}\,\text{m}

    **Final answer: 30+42+213\boxed{30 + 4\sqrt{2} + 2\sqrt{13}} meters}

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