Let w1 = w3 = w5 =w6 = q for some value q. Then w2 = 3q and w4 = 12q. Since the sum of the weights (actually probabilities of coming up) must be 1, we conclude that 19 q = 1 or q = 1/19 Thus w3 = 1/19, ...
Let w1 = w3 = w5 =w6 = q for some value q. Then w2 = 3q and w4 = 12q. Since the sum of the weights (actually probabilities of coming up) must be 1, we conclude that 19 q = 1 or q = 1/19 Thus w3 = 1/19, ...
The key to problems like this one is the triangle inequality: the longest side must be strictly less (in length) than the sum of the lengths of the other two sides. The formal statement is in terms of an inequality, but the bounding values for the length of the third side...
The function f(x) = sin(x) - ln(2x) is the difference of two functions (namely sin(x) and ln(2x) ) The derivative of a difference is the difference of the derivatives. The derivative of sin(x) is cos(x) The derivative...
This problem is of the type that can be solved with the aid of the binomial distribution. Since the probability of getting a head on a single flip is 1/2 as is the probability of getting a tail, the binomial distribution gives the desired probability as ...
The key to story problems like this is that rates add. Let G = Genny's rate and T = Ty's rate. These rates can be thought of as B&Bs cleaned per hour. From the story G + T = 1/( 2 2/9) =...
The answer is d - he will move away from the book. What makes this problem hard to think about is that the mechanics of walking is complicated. To get around this difficulty replace ordinary walking with a series of "bunny hops" in which the student leaps forward, stays...
This can be done for an idea gas if you know the molar mass of the gas. The starting point is the ideal gas equation PV = n RT. Divide both side by P and n to get V/n = R T/ P V/n is a type...
You need the change of base formula. There are various ways to express this formula. Here is one: ln(x) = log10(x) / log10(e) Thus ln(8) = log10(8) / log10(e) ...
Assuming that the second last term is 81 x, this quartic can be factored with aid of a graphing calculator such as the TI-84. Just plot the function and do a little bit of exploration. It is fairly easy to see that there are zeros at x = -4 and x = 3. A little more...
The lens equation can be used to analyze this type of problem (the lens equation works for mirrors also). 1/f = 1/p + 1/q (some books have do in place of p and di in place of q) For the original object distance, ...
This is a classic optimization problem. The formula for the area of a rectangle is A = L W (L= length, W = width) We would like to optimize A, but A depends on two variables not just one. To fix this problem, we use the equation...
I am assuming that the interval that you have in mind is [0,36]. The mean value of the slope is [ f(36) - f(0) ] / 36 = 48/36 = 4/3 . The derivative of f is 4/ sqrt(x) . To find c such that f'(c) = 4/3, set ...
Good job so far. On the left side, notice that (1/3) ln(y) = ln( y1/3). On the right side, notice that the exponential of the sum is the product of (2+x) and ec . But since ec is a constant, it can be replaced with another constant - say C. Then ...
Since both quadratic terms are positive, it is an ellipse. The center is at (2,-3). The major axis is parallel to the y axis with the equation x = 2. Thus the x coordinates of the vertices and foci are all equal to 2. The length of the semi-major...
Assume that: m is the mass of the dart and M is the mass of the wheel, R is the radius of the wheel and v0 is the velocity of the dart just before it strikes and sticks into the edge of the wheel. Assume that the wheel starts at rest. Further assume that the rotation axis of the wheel...
For this reaction, the forward direction is strongly favored and O2 is the limiting reactant. After equilibrium is reached, very little O2 will remain, and the concentration of methane will be approximately .5 - ½.4 = .3, the concentration of CO2 will approximately...
This problem is a bit easier to think about if the roles of x and y are reversed. That is - consider the problem to find the centroid of the plane region bounded by: the parabola y = (1/4) x2 ; the x axis ; and the line x =4. The coordinates...
Let the height of the bottle be h and the area of the bottom be A. In the analysis the area A can be taken as 1 without loss of generality, so I will take A =1. The volume of the bottle is then h. Before the separation, the center of mass...
This problem can be broken down into two parts: First part: The explosive ejection of the cork. The principal of conservation of momentum applies to this event. Before the ejection, the system momentum is zero, so it must also be zero just afterwards...
This problem can be solved if an assumption is made about the angle of repose for a sand pile. According to Wikipedia, the angle of repose for dry sand is ~ 34°. A little bit of trig shows that with this angle, the relationship between the radius of the circular base,r, and...