Let L be the circle in the x-y plane with center the origin and radius 38. Let S be a moveable circle with radius 8 . S is rolled along the inside of L without slipping while L remains fixed. A point P is marked on S before S is rolled and the path of P is studied. The initial position of P is (38,0). The initial position of the center of S is (14,0) . After S has moved counterclockwise about the origin through an angle t the position of P is: x = 14cost + 24cos(7/12t)
y= 14sint - 24sin (7/12t)

Required:
How far does P move before it returns to its initial position?

P moves = 70.73 m

Step-by-step explanation:

Given data

Radius = 38

initial position of P = ( 38,0 )

initial position of center S = ( 14,0)

position of P ( after s moved counterclockwise )

:  x = 14cost + 24cos(7/12t)

  y = 14sint - 24sin(7/12t)

Determine how far P moves before returning to its initial position

attached below  is the solution

P moves = 70.74 m

Job B pays more after taxes.

Explanation:

Job A has taxable income of $8150. This amount is in the interval of $0 to $8350, so we use the first tax bracket.

To be honest, this chart makes things a bit more complicated than they have to be, but it's basically saying "if you're in the first tax bracket, then you pay 10% of your taxable income".

The amount of tax is

10% of 8150 = 0.10*8150 = 815

Which means that he'll have 8150-815 = 7335 dollars after tax.

Now let's move onto job B.

The taxable income of 8650 is now between 8350 and 33950. So we've bumped up to the second tax bracket.

The rule here is that we start with the tax amount being $835. Then we'll add on an extra amount. That extra amount is 15% of anything over 8350. Note how 8650-8350 = 300. So we'll add on 15% of 300 = 0.15*300 = 45 extra dollars on top of the $835 base amount for this bracket.

In total, you'll pay a total of 835+45 = 880 dollars in taxes if you go with job B. The after tax pay is 8650-880 = 7770

To summarize, we have the following after-tax amounts

The difference in the after-tax pay is 7770-7335 = 435 with job B paying more.

Side note: The use of the phrase "taxable income" is different from your actual gross income. This is because you can claim deductions to lower the taxable income. So for instance, Job A could pay an annual salary of $10,000 a year, but through various deductions, he was able to get it down to $8150. The lower your taxable income means you pay less in overall tax (but it doesn't mean your actual income has gone down).