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Probability (answer)

This is just P(X ≤ 25) where X ~ Bin(n=50,p=0.65)   Using R by typing pbinom(25,size=50,prob=0.65), we have:   P(X ≤ 25) = 0.02066839   We can also give a normal approximation   Let Y ∼ N(μ,σ2) where   μ = np =...

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Multiply by √x to get x2 - 14x + 49=0.   This new equation then factors as (x-7)2 = 0   Therefore the solution is x=7   You can easily verify that this solution is not extraneous.

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Recall that the formula for the angle θ between two vectors in U,V ∈ Rn is cos θ = (U·V)/(||U||*||V||)   You probably made a typo when writing what V is.   For U = 〈√5,-8〉 and V = 〈5,1〉, we have   U·V = -8+5√5 ||U|| = √(5+64) = √69 |...

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I think you missed the minus sign on the right hand side. It should be (14x+3) y' - y'' = -14y   Also, the H should be Euler's number "e".   Solution: Just plug it in.   y = e7x^2+3x +2 ⇒ y' = (14x+3) e7x^2+3x +2   ⇒...

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slope intercept (answer)

Part 1: (0,3) and (5,-1):   m = (y2 - y1) / (x2 - x1) = (-1 - 3) / (5 - 0) = -4/5   b = 3 because of the (0,3) point being the site of the y-intercept.   Thus y = (-4/5)x + 3.     Part 2: (3,-5) and parallel to y= -x - 5.   m...

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Note that a full binary tree of height h has exactly 2h leaves.   Proof:   Base: or h=0, the full tree has just the root node, which is a leaf. This is 20 leaves.   Induction: Assume for some k≥0, that the height h full tree has 2h leaves when h=k. Then...

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Recall that the work done against uniform gravity when lifting is W = ∫ F dh = ∫ mg dh where the potential energy at ground level is taken to be 0.   Note how the mass of the bucket at height h is   m(h) = (75 + 500 - 2h)  kg = (575...

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d/dx x2/x   = d/dx exp(ln x2/x)   = d/dx e(2 ln x)/x   = 2e(ln x)/x d/dx [(ln x)/x]   = 2x2/x [(1/x)x - (ln x)1]/x2   = 2x2/x - 2 (1 - ln x)   This function is related to an interesting function f(x) = x1/x for which I...

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The only way this can be elementary integral is if the function is   f(x) = x1/3 + 5/x6   since for all other combinations, Wolfram|Alpha gives an answer in terms of the Hypergeometric function 2F1(a,b;c;x).   ∫ (x1/3 + 5/x6)∫ (x1/3 + 5x-6) dx...

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Word problem (answer)

Let I be his age at inauguration, and D be the age of death.   The first clue is 2I - D = 45.   You get:   D = 2I - 45 = 2·69 - 45 = 138 - 45 = 93   So he died at age 93.   This is indeed his age of death: https://en.wikipedia...

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Find the slope? (answer)

Recall the slope formula: m = (y2 - y1) / (x2 - x1)   (a) m = (-2 - 1) / (3 - -2) = -3/5   (b) m = (-1 - -2) / (5 - 0) = 1/5

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We first need the mean μ and standard deviation σ. We also know that since we are testing at significance α=1-0.95 =0.05. Thus we have: n = 12 xbar = Σxi / n = (4045 + ... + 4324) / 12 = 4341.25 s = √ [Σ(xi - xbar)2 / (n-1)] = √{ [(4045 -...

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