Assume that the following confidence interval for the difference in the mean time (in minutes) for male students to complete a
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Answer:
α ≥ 48.2°
Step-by-step explanation:
The complete question is given as follows:
" A skier starts at the top of a very large frictionless snowball, with a very small initial speed, and skis straight down the side. At what point does she lose contact with the snowball and fly off at a tangent? That is, at the instant she loses contact with the snowball, what angle α does a radial line from the center of the snowball to the skier make with the vertical?"
- The figure is also attached.
Solution:
- The skier has a mass (m) and the snowball’s radius (r).
- Choose the center of the snowball to be the zero of gravitational potential. - We can look at the velocity (v) as a function of the angle (α) and find the specific α at which the skier lifts off and departs from the snowball.
- If we ignore snow-ski friction along with air resistance, then the one work producing force in this problem, gravity, is conservative. Therefore the skier’s total mechanical energy at any angle α is the same as her total mechanical energy at the top of the snowball.
- Hence, From conservation of energy we have:
KE (α) + PE(α) = KE(α = 0) + PE(α = 0)
0.2*m*v(α)^2 + m*g*r*cos(α) = 0.5*m*[ v(α = 0)]^2 + m*g*r
0.2*m*v(α)^2 + m*g*r*cos(α) ≈ m*g*r
m*v(α)^2 / r = 2*m*g( 1 - cos(α) )
- The centripetal force (due to gravity) will be mgcosα, so the skier will remain on the snowball as long as gravity can hold her to that path, i.e. as long as:
m*g*cos(α) ≥ 2*m*g( 1 - cos(α) )
- Any radial gravitational force beyond what is necessary for the circular motion will be balanced by the normal force—or else the skier will sink into the snowball.
- The expression for α_lift becomes:
3*cos(α) ≥ 2
α ≥ arc cos ( 2/3) ≥ 48.2°
Given:
Wang Yong owes the bank $8500. He pays back a fixed amount each month. After 12 months, he still has to pay off $6460.
To Find:
The amount of money he pays the bank every month and how long it will take him to pay off the full debt.
Answer:
Wang Yong pays $170 each month, and doing so, it will take him a total time of 50 months to pay off the full debt.
Step-by-step explanation:
Wang Yong has borrows $8500 according to the question.
Say he pays the bank an amount of x every month, which is fixed. After 12 months, he still has to pay $6460.
In 12 months, he would have paid the bank 12x.
But $6460 is still left out of the original $8500.
This means,
That is, Wang Yong pays the bank $170 every month.
Next, to calculate the amount of time it took Wang Yong to pay back his whole debt, we divide the total sum of money owed to the bank by the amount Wang Yong pays each month.
So, number of months = 8500 ÷ 170 = 50
That is, Wang Yong will pay off his debt 50 months after borrowing the money from the bank, provided he pays back $170 every month.