Answer:
The selling price that will maximize profit is $56.
Step-by-step explanation:
Given : It costs 12 dollars to manufacture and distribute a backpack. If the backpacks sell at x dollars each, the number sold, n, is given by ![n=\frac{2}{x-12}+7(100-x)](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B2%7D%7Bx-12%7D%2B7%28100-x%29)
To find : The selling price that will maximize profit ?
Solution :
The cost price is $12.
The selling price is $x
Profit = SP-CP
Profit = x-12
The profit of n number is given by,
![P=(x-12)n](https://tex.z-dn.net/?f=P%3D%28x-12%29n)
Substitute the value of n,
![P=(x-12)(\frac{2}{x-12}+7(100-x))](https://tex.z-dn.net/?f=P%3D%28x-12%29%28%5Cfrac%7B2%7D%7Bx-12%7D%2B7%28100-x%29%29)
![P=\frac{2(x-12)}{x-12}+7(100-x)(x-12)](https://tex.z-dn.net/?f=P%3D%5Cfrac%7B2%28x-12%29%7D%7Bx-12%7D%2B7%28100-x%29%28x-12%29)
![P=2+7(100x-1200-x^2+12x)](https://tex.z-dn.net/?f=P%3D2%2B7%28100x-1200-x%5E2%2B12x%29)
![P=2+700x-8400-7x^2+84x](https://tex.z-dn.net/?f=P%3D2%2B700x-8400-7x%5E2%2B84x)
![P=-7x^2+784x-8398](https://tex.z-dn.net/?f=P%3D-7x%5E2%2B784x-8398)
Derivate w.r.t x,
![\frac{dP}{dx}=-14x+784](https://tex.z-dn.net/?f=%5Cfrac%7BdP%7D%7Bdx%7D%3D-14x%2B784)
Put it to zero for critical point,
![-14x+784=0](https://tex.z-dn.net/?f=-14x%2B784%3D0)
![-14x=-784](https://tex.z-dn.net/?f=-14x%3D-784)
![x=\frac{-784}{-14}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-784%7D%7B-14%7D)
![x=56](https://tex.z-dn.net/?f=x%3D56)
Derivate again w.r.t x, to determine maxima and minima,
![\frac{d^2P}{dx^2}=-14](https://tex.z-dn.net/?f=%5Cfrac%7Bd%5E2P%7D%7Bdx%5E2%7D%3D-14%3C0)
It is a maximum point.
Therefore, the selling price that will maximize profit is $56.
13% is bigger than 3/25. If you turn the 13% into a fraction it comes out to be 13/100.
To compare the two they have to have he same denominator. To make 3/25 have the denominator of 100 multiply the numerator and denominator by 4. You should get 12/100.
12/100<13/100
By Green's theorem,
![\displaystyle\int_C\cos y\,\mathrm dx+x^2\sin y\,\mathrm dy=\iint_D\left(\frac{\partial(x^2\sin y)}{\partial x}-\frac{\partial(\cos y)}{\partial y}\right)\,\mathrm dx\,\mathrm dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_C%5Ccos%20y%5C%2C%5Cmathrm%20dx%2Bx%5E2%5Csin%20y%5C%2C%5Cmathrm%20dy%3D%5Ciint_D%5Cleft%28%5Cfrac%7B%5Cpartial%28x%5E2%5Csin%20y%29%7D%7B%5Cpartial%20x%7D-%5Cfrac%7B%5Cpartial%28%5Ccos%20y%29%7D%7B%5Cpartial%20y%7D%5Cright%29%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy)
where
is the region with boundary
, so we have
![\displaystyle\iint_D(2x+1)\sin y\,\mathrm dx\,\mathrm dy=\int_0^5\int_0^4(2x+1)\sin y\,\mathrm dy\,\mathrm dx=\boxed{60\sin^22}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_D%282x%2B1%29%5Csin%20y%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%3D%5Cint_0%5E5%5Cint_0%5E4%282x%2B1%29%5Csin%20y%5C%2C%5Cmathrm%20dy%5C%2C%5Cmathrm%20dx%3D%5Cboxed%7B60%5Csin%5E22%7D)
Radius is 1/2 the circumference so 2.8 x 2= 5.6 ft. (Ceiling fan)
13 x 2 = 26mm (water bottle cap)
56.5•2= 28.25cm (bowl)
75.4•2= 37.7 in (drum)
Answer:
He is expected to win 25% of the time
50 times
Step-by-step explanation:
If we use fair coins
The probability for one coin landing face up is 0.5
(50% of the time)
Since we need both coins landing face up, we multiply the probabilities
(0.5)*(0.5) = 0.25
(25 % of the time)
This means that
0.25 * 200 times = 50 times