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Express the vector BC in terms of the given vectors AB and AC.

Mathematics

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Answer

(40)\begin{pmatrix} 4 \\ 0 \end{pmatrix}

615.75 cm2615.75 \text{ cm}^2

Step-by-step solution

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let's solve these two problems.

: Vector Midpoint

  1. Express the vector BC⃗\vec{BC} in terms of the given vectors AB⃗\vec{AB} and AC⃗\vec{AC}.

    We know that BC⃗=BA⃗+AC⃗\vec{BC} = \vec{BA} + \vec{AC}.
    Since BA⃗=−AB⃗\vec{BA} = -\vec{AB}, we have:

    BC⃗=−AB⃗+AC⃗\vec{BC} = -\vec{AB} + \vec{AC}

    Substitute the given vectors:

    BC⃗=−(64)+(2−4)\vec{BC} = -\begin{pmatrix} 6 \\ 4 \end{pmatrix} + \begin{pmatrix} 2 \\ -4 \end{pmatrix} BC⃗=(−6−4)+(2−4)\vec{BC} = \begin{pmatrix} -6 \\ -4 \end{pmatrix} + \begin{pmatrix} 2 \\ -4 \end{pmatrix} BC⃗=(−6+2−4−4)\vec{BC} = \begin{pmatrix} -6 + 2 \\ -4 - 4 \end{pmatrix} BC⃗=(−4−8)\vec{BC} = \begin{pmatrix} -4 \\ -8 \end{pmatrix}

  2. Calculate the position vector of the midpoint M of BC relative to point A.

    Let M be the midpoint of BC. The position vector of M relative to A, denoted AM⃗\vec{AM}, can be found using the formula:

    AM⃗=12(AB⃗+AC⃗)\vec{AM} = \frac{1}{2}(\vec{AB} + \vec{AC})

    Substitute the given vectors:

    AM⃗=12((64)+(2−4))\vec{AM} = \frac{1}{2} \left( \begin{pmatrix} 6 \\ 4 \end{pmatrix} + \begin{pmatrix} 2 \\ -4 \end{pmatrix} \right) AM⃗=12(6+24−4)\vec{AM} = \frac{1}{2} \begin{pmatrix} 6 + 2 \\ 4 - 4 \end{pmatrix} AM⃗=12(80)\vec{AM} = \frac{1}{2} \begin{pmatrix} 8 \\ 0 \end{pmatrix} AM⃗=(40)\vec{AM} = \begin{pmatrix} 4 \\ 0 \end{pmatrix}

    The midpoint of BC⃗\vec{BC} (represented as the position vector from A) is (40)\boxed{\begin{pmatrix} 4 \\ 0 \end{pmatrix}}.

    : Total Surface Area of a Cone

  3. Identify the given values for the cone.

    Radius of the base, r=OC=7r = OC = 7 cm.
    Slanting height, l=AC=21l = AC = 21 cm.

  4. Recall the formula for the total surface area of a cone.

    The total surface area (TSATSA) of a cone is the sum of the area of its circular base and its curved surface area.

    TSA=πr2+πrlTSA = \pi r^2 + \pi r l

    This formula can also be written as:

    TSA=πr(r+l)TSA = \pi r (r + l)

  5. Substitute the given values into the formula and calculate the total surface area.

    TSA=π(7 cm)(7 cm+21 cm)TSA = \pi (7 \text{ cm}) (7 \text{ cm} + 21 \text{ cm}) TSA=π(7 cm)(28 cm)TSA = \pi (7 \text{ cm}) (28 \text{ cm}) TSA=196π cm2TSA = 196\pi \text{ cm}^2

    To get a numerical value, use π≈3.14159\pi \approx 3.14159:

    TSA≈196×3.14159TSA \approx 196 \times 3.14159 TSA≈615.75244 cm2TSA \approx 615.75244 \text{ cm}^2

    Rounding to two decimal places:

    TSA≈615.75 cm2TSA \approx 615.75 \text{ cm}^2

    The total surface area of the cone is 615.75 cm2\boxed{615.75 \text{ cm}^2}.

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