Search 75,706 tutors

# Which is the first number in this series which is less than 0.01? (answer)

This problem can be solved by trial and error (guess and check).   The analytic approach is to note that the general term in the sequence is:   an  =  25-n   So we want     25-n  < .01   We can take log (base...

# sin(x/2) + cos(x/2) = ? (answer)

You need the half angle formulas.   sin(x/2) = sqrt( (1-cos(x) ) /2 )   cos(x/2) = sqrt( (1 + cos(x) )/2)   There can be a ± in front of the square roots in some situations.

This problem can be solved very easily with calculus.     At the maximum value of θ ,  the derivative of the expression will be zero.    This derivative is    12 cos(θ) - 18 sin(θ) cos(θ)   setting this to zero results in 12...

The polynomial remainder theorem states that if a polynomial , f(x),  is divided by (x -a) using long division of polynomials, the remainder will be f(a).    For example  if the polynomial is   f(x) = x2 -9      and we divide by x-2,  ...

# help with voltage and electricity c/o physical science (answer)

The first statement is essentially true.   The conduction band electrons in single domain of iron do have parallel spins.  Since the electron has a magnetic moment, a magnetic field results.   The second is almost true. A voltage will be induced in a wire coil moving...

A special solution can be fairly easily guessed.   It is   f =  (1/(w -k)) cos(kx -wt) + C.   To this we would have to add a general solution to the homogenous equation  df/dx + df/dt = 0   This general solution looks like  ...

# The missing number in the series 9, ____, 6561, 43046721 is: (answer)

Well the sequence is   91  92  94  98   So the second term is 81.

# Find all the zeros of the function f(g) = g^4 + 2g^3 + 10g^2 + 18g + 9 (answer)

The rational roots theorem says that the possible rational roots of this quartic are +1, -1, + 3, -3, +9, -9. Trying all six gets just one hit  g = -1   This means that the quartic is a product of (g+1) and a cubic polynomial.  The cubic polynomial can...

# sin a+cos a+tan a+sec a+cosec a+cot a=7.then prove that sin 2a is the root of x^2-44x+36 (answer)

Using standard trig identities, and multiplying through by sin a cos a, the starting equation can be rewritten as    (sin a  +  cos a) (1 + sin a  cos a ) =  7 sin a  cos a   - 1   Next, we replace  sin a cos a ...

# Select the x-coordinate of the vertex of the parabola defined by the function f(x) = -9x2 + 5x + 9. (answer)

The standard form of the equation of a parabola is   f(x)  = a x2 + b x + c.   With this standard from, the x coordinate of the vertex  is   -b/(2a).   Comparing with the function given, we see that a = -9    ...

# express matrix A as a skew symmetric and symmetric matrix (answer)

If A is a general square matrix, it can be written as   A =  S  + V     where   S ≡  (AT + A) /2   and   V  ≡  (-AT + A) /2   (T denotes transpose)   with these...

# What are the dimensions of such a rectangle with the greatest possible area? (answer)

This problem can be set up as an optimization problem in calculus   The points on the parabolic curve can be written as (x, 11- x2) so the upper right corner will have the coordinates of (x, 11-x2) for some optimal value of x   The width = 2x   and the height...

# What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? (answer)

This problem can also be solved using calculus techniques.   We need a figure similar to the one described in the non-calculus answer.   If we call h the height above the ground at which the ladder touches the building, and x the distance from the foot of the ladder to the...

Rather than track your steps, I will show you how I would approach this problem.  Areas can be computed as integrals over either x or y.    For the problem at hand, it is easier to think of x as function of y and to carry out the integral over y.  Notice that the equations...

# What is the apparent power of an 8A, 120VRMS motor with a power factor of 0.90? (answer)

Power is normally voltage (RMS in the case of AC) times current (also RMS in the case of AC).  So   P = 120 x 8 x .9  =   864 Watt   The 0.9 comes in because (again for AC) the voltage and current are not fully in phase.

Since the cdf is a smooth function for 0 < X < 1,   P( X < .8)  is just cdf(.8)  =   .64

# At what rate in cubic centimeters per minute is the volume increasing at this instant? (answer)

This problem can be solved by differentiation, with respect to time, of the equation      P V1.4  = C   The derivative of the right hand side is zero, and the derivative of the left hand side can be worked out using the product rule and the power rule...