The reaction shown is a reduction of an aldehyde to an alcohol. This is typically done with a reducing agent such as
NaBH4. The aldehyde is reduced, but the atom within the aldehyde that is gaining the electrons (being reduced) is usually thought of as the being the carbon...

The expression on the left hand side can be factored so:
(2 cos(θ) -1) ( 2 cos(θ) +3) = 0
Thus either 2 cos(θ) -1 =0 or 2 cos(θ) +3 =0
but there are no solutions for 2 cos(θ)...

The standard method for determining the empirical formula given the concentration by mass is the following:
1) consider a sample of 100 g. Then work out the mass of each element. For this problem, the mass of C would be 24.24 g, for H, 4.04 g, for Cl, 71.72...

Call the center of the circle point F. Then m∠HFJ = 180 - 101 = 79. The angle HFJ is a central angle. Therefore the minor arc HJ also has measure 79. The major arc HJ has measure 360 - 79 = 281.

The apothem, a, of a regular pentagon is the distance from the center of the pentagon to the midpoint of an edge.
Trigonometry can be used to show that a = (s/2) /tan(pi/10), where s is the length of a side. This is an exact result.
The area, A, of...

The simplest way to set this up is to note that 1 - .028 = .972 So
A(t) = A(0) .972t Where t is number of hours.
Since the formula listed is in e format this equation can be rewritten as
A(t) = A(0) exp [ ln(...

There are two possible answers depending on how the 9 ft is interpreted. If the 9 ft is the distance along the ground between Charles and the mirror, the calculation is:
6/9 = h/63 and h = 42.
If the 9 ft is the...

This is a fairly interesting problem with several elements.
The first is to note that 8 cos(θ) - 6 sin(θ) is equal to 10 cos(θ + arctan(3/4) ) .
This is a standard transformation that can be checked by using the cosine of a sum formula.
The arctan(3/4) is just a phase shift...

The maximum possible volume would be for the case that the rectangular prism is a cube. In that case each of the six sides would have an area of 724 / 6 = 120.66 cm2 . The edge would be sqrt(120.66) = 10.98 cm. The volume would be the cube of this: V = 1325...

The method of Lagrange multipliers can be used to find extrema lying in the interior of some domain.
It appears that for this problem, the domain is the first octant where x,y,z are non negative.
A straightforward application of the Lagrange method finds the value 1/27 at...

39.7 g of N2 is 39.7 / 28 = 1.418 moles of N2 . (This is because the molar mass of N2 = 28 g)
The chemical equation for this reaction (called the Haber reaction) is:
N2 + 3 H2 ==> 2 NH3
...

Angular momentum will be conserved in this inelastic collision.
So Li = 161 = Lf
Lf = I ω , where I is the moment of inertia of the disk+child system and ω is the desired angular speed.
...

The slope of the line CD is (12 - 11)/(5 - 0) = 1/5
The slope of the perpendicular bisector is the negative reciprocal of this (that is -5)
The midpoint of line CD is (5/2 , 23/2)
The point-slope formula can be used to get...

My assumption is that equation is :
(x2 +20)1/2 = 3 z1/2
The first step is to square both sides:
x2 + 20 = 9 x Rearrange this as"
...

Since point A and point B have the same y coordinate, this line is horizontal. The point P will also have y coordinate of 2. The remaining problem is to find the x coordinate of P.
One way to do this is as follows.
For some...

There are standard formulas for this type of problem. However, it can also be solved by a combination of simple steps.
First, the shortest distance from the point (3,5) to the line y = x +4 (line1) will be along a straight line perpendicular to line1. ...

This problem can be solved using the principle that external work done on a system is equal to the change in energy of the system. In symbols
Wext = ΔE = Ef - Ei (here E is KE +PE)
I am going to assume...

The rotational kinetic energy, Krot, is (1/2) I ω2
For a disk I = (1/2) m R2
For the case of rolling without frictions ω = vcm/R
Putting this together Krot = (1/4) m vcm2 This is one half the translational...

One way to solve this kind of problem is to use a description of lines drawn from formal linear algebra.
The line (which will be referred to as Line1): y = x +7 can be described by the ordered triplet (x,y,z)
(0, 7, 0) + α (1 , 1 ,0) = ...

It looks like you are on the right track with this one. However, you have a factor of 2 in your equations that I don't understand.
Since ρ is quoted in the SI unit (kg/m3) and ultimately one would like to used g = 9.8 m/s2 , it is a good idea to convert the...