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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Step 1: Find the larger value of the fuel flow rate x using Model 1. Model 1: _2 x + 4 _x 8 = 8 First, change the base of _x 8 to base 2 using the change of base formula _b a = (_c a)/(_c b): _x 8 = (_2 8)/(_2 x) = (3)/(_2 x) Substitute this into the equation: _2 x + 4 ((3)/(_2 x)) = 8 _2 x + (12)/(_2 x) = 8 Let u = _2 x. The equation becomes: u + (12)/(u) = 8 Multiply by u: u^2 + 12 = 8u Rearrange into a quadratic equation: u^2 - 8u + 12 = 0 Factor the quadratic equation: (u-2)(u-6) = 0 This gives two possible values for u: u=2 or u=6. Substitute back u = _2 x: If u=2, then _2 x = 2 x = 2^2 = 4. If u=6, then _2 x = 6 x = 2^6 = 64. The larger value of x is 64. Step 2: Find the thrust coefficient k and the final thrust control T using Model 2. Model 2: 3^2k+1 - 28(3^k) + 9 = 0 Rewrite 3^2k+1 as 3^2k · 3^1 = 3 · (3^k)^2: 3 · (3^k)^2 - 28(3^k) + 9 = 0 Let v = 3^k. The equation becomes a quadratic equation in terms of v: 3v^2 - 28v + 9 = 0 Factor the quadratic equation: (3v-1)(v-9) = 0 This gives two possible values for v: v=(1)/(3) or v=9. Substitute back v = 3^k: If v=(1)/(3), then 3^k = (1)/(3) 3^k = 3^-1 k=-1. If v=9, then 3^k = 9 3^k = 3^2 k=2. The problem states to use the larger value of k, so k=2. Now, calculate the final thrust control T using T = 200k: T = 200 · 2 T = 400 The thrust coefficient k is 2, and the final thrust control T is 400. Step 3: Find the value of y using Model 3. Model 3: sqrt(y+5) - sqrt(y-2) = 1 Isolate one square root term: sqrt(y+5) = 1 + sqrt(y-2) Square both sides of the equation: (sqrt(y+5))^2 = (1 + sqrt(y-2))^2 y+5 = 1 + 2sqrt(y-2) + (y-2) y+5 = y - 1 + 2sqrt(y-2) Subtract y from both sides: 5 = -1 + 2sqrt(y-2) Add 1 to both sides: 6 = 2sqrt(y-2) Divide by 2: 3 = sqrt(y-2) Square both sides again: 3^2 = (sqrt(y-2))^2 9 = y-2 y = 9+2 y = 11 Check the solution: sqrt(11+5) - sqrt(11-2) = sqrt(16) - sqrt(9) = 4 - 3 = 1. The solution is valid. The value of y is 11. Step 4: Show that the sum of the results (x+k+y) satisfies the relationship _2(x+k+y-13) = 6. First, calculate the sum x+k+y: x+k+y = 64 + 2 + 11 = 77 Now substitute this sum into the given relationship: _2(77 - 13) = 6 _2(64) = 6 Since 2^6 = 64, the statement _2(64) = 6 is true. Thus, the sum of the results (x+k+y) satisfies the given relationship. The final answers are: a) (i) The larger value of the fuel flow rate x is 64. a) (ii) The thrust coefficient k is 2 and the final thrust control T is 400. a) (iii) The value of y is 11. b) The sum x+k+y = 64+2+11 = 77. Substituting into the relationship: _2(77-13) = _2(64) = 6. This shows the relationship is satisfied. That's 2 down. 3 left today — send the next one.