This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — let's solve these two problems. Problem 8: Vector Midpoint Step 1: Express the vector BC in terms of the given vectors AB and AC. We know that BC = BA + AC. Since BA = -AB, we have: BC = -AB + AC Substitute the given vectors: BC = - 6 \\ 4 + 2 \\ -4 BC = -6 \\ -4 + 2 \\ -4 BC = -6 + 2 \\ -4 - 4 BC = -4 \\ -8 Step 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 AM, can be found using the formula: AM = (1)/(2)(AB + AC) Substitute the given vectors: AM = (1)/(2) ( 6 \\ 4 + 2 \\ -4 ) AM = (1)/(2) 6 + 2 \\ 4 - 4 AM = (1)/(2) 8 \\ 0 AM = 4 \\ 0 The midpoint of BC (represented as the position vector from A) is 4 \\ 0 . Problem 9: Total Surface Area of a Cone Step 1: Identify the given values for the cone. Radius of the base, r = OC = 7 cm. Slanting height, l = AC = 21 cm. Step 2: Recall the formula for the total surface area of a cone. The total surface area (TSA) of a cone is the sum of the area of its circular base and its curved surface area. TSA = r^2 + r l This formula can also be written as: TSA = r (r + l) Step 3: Substitute the given values into the formula and calculate the total surface area. TSA = (7 cm) (7 cm + 21 cm) TSA = (7 cm) (28 cm) TSA = 196 cm^2 To get a numerical value, use ≈ 3.14159: TSA ≈ 196 × 3.14159 TSA ≈ 615.75244 cm^2 Rounding to two decimal places: TSA ≈ 615.75 cm^2 The total surface area of the cone is 615.75 cm^2. That's all the problems from the image! What's next?