The expression can be factored giving: (sin(x) +4)(4 sin(x) - 1) = 0 Since sin(x) cannot equal 4, this implies 4 sin(x) -1 = 0. Thus sin(x) = 1/4 x = arcsin(1/4) (1st quadrant) x...

4sin2(x) + 15sin(x) - 4 = 0 (answer)

The expression can be factored giving: (sin(x) +4)(4 sin(x) - 1) = 0 Since sin(x) cannot equal 4, this implies 4 sin(x) -1 = 0. Thus sin(x) = 1/4 x = arcsin(1/4) (1st quadrant) x...

The numerator factors as (sin(a) +6) (sin(a) -6) The denominator factors as (sin(a) + 6 ) (sin(a) +6) On dividing, there is a cancellation of a factor of (sin(a) +6) with the resuilt (sin(a) -6) / (sin(a) +6)

the optimal reception (answer)

The standard form of the equation of parabola with vertex at the origin is y = a x2 In this case y = 8 when x = 36/2 = 18 . Plugging in and solving for a gives a = 0.02469 The P value for a...

Using implicit differentiation, I get dy/dx = -(2 x ey + y ex) /(x2 ey + ex) Evaluating this for x = 2, y = 0 gives m = dy/dx = -4/(4 + e2) Using the point slope form, the equation...

The yield per tree depends on the number of trees. Let y(N) be the yield per tree if N trees are planted in the lot. The formula for y(N) can be seen to be y(N) = 1500 - 20 (N-65). The size of the crop, S, is given by: S(N)...

for 0 < θ < π/2 , these curves cross at θ =arctan( 31/2 ) = π/3 The cosine curve is the upper one, so the desired area is A = integral from 0 to π/3 of [ 31/2 cos(θ) - sin(θ) ] The anti-derivative...

The law of cosines can be used as the input to the process of implicit differentiation. Let D be the distance between the friends and θ be the angle between (the line joining the center of the circle and the standing friend ) and (the radius line joining the center of the circle and...

One way to analyze this problem is to change from a rectangular system (x,y) to a polar system (r , θ) via x = r cos(θ) , y = r sin(θ). The expression to be analyzed becomes: [r3 cos3(θ) + r3 sin3(θ) ]/r2 = r[cos3(θ)...

Without drag, the time up equals the time down. With drag, the average speed on the way up is greater than the average speed on the way down. Since the distance covered is the same on the way up as it is on the way down, the trip down must take longer. A simple...

This type of problem can be modeled as a Poisson process. If the player takes 20 shots in a game, the expected number of shots he will hit is .46 x 20 = 9.2. This number becomes λ in a Poisson model. For a Poisson process, probability of hitting...

Limits...is my answer correct? (answer)

Yes. In fact this one can be done by direct substitution. cot(pi/2) = 0 and csc(pi/2) =1 0/1 is not an indeterminate form , so the answer is 0.

There are four jacks in standard deck of cards (spade, heart, diamond, club). Thus the number ways that you can be dealt two jacks is the combinatorial number 4C2 - commonly called (four choose two) This number is 4!/(2! 2!) = 6. The number of ways that you...

What is x equal to in this equation (answer)

One way to address this problem is to ask: What is the lower limit, as x approaches zero from above, of sqrt(x + sqrt(x + sqrt(x + sqrt(x + ...) )) I do not know how to rigorously evaluate this limit. However,...

What is the probability of you getting something that only has a 5% chance if you do it 14 times? (answer)

The probability of not getting it in 14 tries is (0.95)14 So the probability of getting it at least once is 1 - (0.95)14 ~ 1 - .488 ~ .512 So in 14 tries there is a better than even chance of getting the desired...

Find angle. (answer)

I will assume that point E is inside the square. The length of diagonal AC is 10 sqrt(2) The law of sines gives: sin(154) / (10 sqrt(2)) = sin( ∠EAC)/ 6. Solving for ∠ EACV gives ...

Complete combustion implies that the products are carbon dioxide and water. So the pure gas is CO2 . There was 40 L of CO2 at STP. Dividing by the standard molar volume (22.4 L) gives 1.786 moles of CO2 . The mass of water was 56.35 g. Dividing by the molar...

how to factor a^3b-ab^3+a^2+b^2+1 (answer)

The answer to this one is fairly simple, but the reasoning leading to the answer is not. The factorization is [ (a2+b2)/2 +ab- (a2-b2)/2 +1] [ (a2+b2)/2 -ab +(a2-b2) +1] This can be checked by multiplying it all out. The...

Find the equation of an ellipse. (answer)

Since the length of the major axis is 10, the semi-major axis, a, is a = 5 The distance from the center to a focus is the c value, c = sqrt( a2 - b2) where b is the semi-minor axis. Since c = 2, b = sqrt(21) The...

if y = ux, then y' = x u' + u and the diff equation is x u' + u = f(u) x u' = f(u) -u dx / x = du /( f(u) -u ) so ln...

There are at least two ways to solve this triangle problem. One method (used here) is trigonometric / classical geometric Bearings are measured from the North direction. Thus the direction from A to B is rotated 78 degrees away from North. The...

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