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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Welcome back ♡tasha♡....🧸 — missed you this week. Here are the solutions to the problems. --- Question 6: A bag contains 3 different marbles x, y and z. The total number of marbles in the bag is 27. The ratios are x:y = 1:2 and y:z = 1:3. a) Find the number of z marbles in the bag: Step 1: Combine the ratios to find x:y:z. Given x:y = 1:2. Given y:z = 1:3. To combine these, we make the y component consistent. We can multiply the second ratio by 2: y:z = (1 × 2) : (3 × 2) = 2:6. Now, we have x:y = 1:2 and y:z = 2:6. So, the combined ratio is x:y:z = 1:2:6. Step 2: Calculate the total number of parts in the ratio. Total parts = 1 + 2 + 6 = 9 parts. Step 3: Determine the value of one part. The total number of marbles is 27. Value of one part = Total marblesTotal parts = (27)/(9) = 3 marbles per part. Step 4: Calculate the number of z marbles. The ratio for z is 6 parts. Number of z marbles = 6 × 3 = 18. b) Calculate the probability of selecting two of the y marbles at random from the bag one after the other without replacement. Step 1: Calculate the number of y marbles. From the combined ratio x:y:z = 1:2:6, the ratio for y is 2 parts. Number of y marbles = 2 × 3 = 6. Step 2: Calculate the probability of selecting the first y marble. Total marbles in the bag = 27. Number of y marbles = 6. P(1st y) = Number of y marblesTotal marbles = (6)/(27) Step 3: Calculate the probability of selecting the second y marble without replacement. After selecting one y marble, there are 6-1=5 y marbles left. The total number of marbles remaining in the bag is 27-1=26. P(2nd y | 1st y) = Remaining y marblesRemaining total marbles = (5)/(26) Step 4: Calculate the overall probability. The probability of selecting two y marbles one after the other without replacement is the product of the probabilities from Step 2 and Step 3. P(two y marbles) = P(1st y) × P(2nd y | 1st y) P(two y marbles) = (6)/(27) × (5)/(26) Simplify the fractions: P(two y marbles) = (2)/(9) × (5)/(26) P(two y marbles) = (1)/(9) × (5)/(13) P(two y marbles) = (5)/(117) --- Question 7: Given that m = 2i + pj, n = -3i + 4j and = (6)/(5sqrt(13)), where is an angle between m and n. Find the values of p. Step 1: Recall the dot product formula for two vectors. The dot product of two vectors m and n is given by: m · n = |m| |n| Step 2: Calculate the dot product m · n. Given m = 2i + pj and n = -3i + 4j. m · n = (2)(-3) + (p)(4) = -6 + 4p Step 3: Calculate the magnitudes of vectors |m| and |n|. |m| = sqrt(2^2 + p^2) = sqrt(4 + p^2) |n| = sqrt((-3)^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 Step 4: Substitute the values into the dot product formula. We have m · n = -6 + 4p, |m| = sqrt(4 + p^2), |n| = 5, and = (6)/(5sqrt(13)). -6 + 4p = (sqrt(4 + p^2))(5)((6)/(5sqrt(13))) Step 5: Simplify the equation. -6 + 4p = sqrt(4 + p^2) ((6)/(sqrt(13))) Multiply both sides by sqrt(13): sqrt(13)(-6 + 4p) = 6sqrt(4 + p^2)