This problems is a bit tricky because the resulting solid should be viewed as being into two parts joined together.   The first part corresponds to 0 ≤ x < 1 and the second to   1 ≤ x  ≤ 9   The volume of the first part can be computed...

This is a classic optimization problem in calculus.    Let s be the length of the sides of the squares to be cut out.   Then the dimensions of the open box will be 20 - 2s by 20 -2s by s  . The volume is thus  V  = (20 - 2s)2 s   To find the...

With the addition of the 350 mL of water, the concentration of acetic acid is (0.1) 0.035 /(.035 + .35) = 0.00909   The equilibrium is :     1.8 x 10-5  =  [ H+ ] [ Ac-]/ [H Ac]   but [Ac-] = [ H+]      call this x...

I will use an ordered triplet (  ,   ,  ) notation to describe vectors.   The direction of line l  is specified by (1 , - 1, 2)   and the direction of line my by ( 1 , 2, -2)  A direction perpendicular to both of these is the cross product   (...

The first step is to compute the molar mass of Ag2 CO3  .   Using the periodic table this can be found to be about 275.8 g Thus   8.7 mg  is  0.0000319 moles of   silver carbonate. Since each formula unit has 2 silver atoms,  and one carbonate,...

One method starts with a choice for the total energy , E.    Once E is selected, find a value of x for which V(x)  = E.     (call that value x0 ) Then since p =  ± √2m  sqrt( E - V(x)),    a first point on the phase plot (p,x)  is ...

Assuming that the comparison is between an elastic collision of a moving train with a stationary golf ball and an elastic collision of a moving golf club with a stationary golf ball, the ball would go further when driven by the train if the train and club were moving at the same speed.   The...

For part a)   The x coordinate of the center of mass is      xcm =  ∫∫ dθ r dr  r cos(θ)  /   M    where M = ∫∫ dθ r dr        The y coordinate of the center of mass is    ycm...

Given that tanA =  1/√5  ,  A well known triangle analysis can be used to find cscA and secA.   The result is cscA = √6 and secA = √(6/5)     if A is in the first quadrant and  -1 times these if  A is in the third quadrant   The...

Since the vertex is  (-2,-7)   The vertex form of the quadratic function is:  f(x)  =  a (x+2)2 -7  for some value of a.   The y intercept is -3   means f(0) = -3   so   -3 =  a( 0 +2)2  -7 ...

Rewrite the equation as    y2 (1+ sin2x)2 = cos22x      Then replace cos22x with 1 - sin22x    The result is an equation quadratic in sin2x   The quadratic formula can then be used to get two values for sin2x  One is  sin2x =...

First work out the molar mass of glucose   MM = 6 x 12 + 12 x 1 + 6 x 16 =  180 g   The number of moles of glucose is  2.2 x 10-2 / 160 =  0.0001222 moles of glucose.   The number of molecules is this number multiplied by the Avogadro...

At the top if the curve, the occupants of the car are accelerating downward with ac = v2 /R  where v is the velocity and R = 91 m  ((centripetal acceleration)    The free body diagram for an occupant shows two forces:  1) , the downward force of gravity = mg ...

For the case of an equal mass elastic collision, the angle between the exit velocities is 90 degrees.  Hence second ball moves at an angle 90 - 37 = 53 degrees with respect to the original line of motion.

Let c = the length of string to be used for the circle, and s = the length to be used for the square Then   c + s = 22      This is a constraint equation.   s is actually the perimeter of the square, so the side of square is s/4 and the area is ...

To see what is going on,  work backwards from the answer. Long division (or synthetic division)  of   (-2s^3-8s^2+4s+40)/(s^3+4s^2+3s)   Works out to be -2 with a remainder of (10 s +40) /(s^3 +4 s^2 + 3s)   The -2...

The combined gas law can be used to solve this   P1 V1  / T1  =  P2 V2 /  T2     Plugging in  P1  =18   ; T1 = 273  ; V2 =  2 V1   ;  T2 = 298   and cancelling out the factors of V1   The...

I would say this number belongs to the set of real numbers.

The sides are in the ratio 3:4:5,  which is one of the Pythagorean right triangles, so angle A is 90 degrees.   The area, A, is  A = (1/2) 3  (4) =  6.       The area is also equal to:  (1/2) ( the altitude from A to BC) 5   6=...

The Stoichiometric balanced equation is   NaHCO3 + HCl ==>  CO2 + NaCL + H2O   Thus the molar ratios are all 1:1   The initial pressure in the flask was not specified, nor was the Kelvin temperature.   However, assuming that the...