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3 years ago

7

There are 14 dogs in a particular class at a dog show. Blue, red, yellow, and white ribbons will be awarded to the first through

fourth place finishers, respectively. In how many different ways can the ribbons be awarded?
1001
6006
24024
38416

1 answer:

stira [4]3 years ago

4 0

There are 14 possibilities for first place.
There are 13 possibilities for second place, since a dog can't get both first and second place.
There are 12 possibilities for third place, since a dog can't get both first, second, and third place.
There are 11 possibilities for fourth place, since a dog can't get both first, second, third, and fourth place.

Multiply them together and get 24024 possibilities.

Hope this helps!

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b. Rate at which the revenue changing in the year 2010 is 76 million dollar per year ( decreasing)

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Therefore, there is loss of revenue at the beginning of 2010 which is 75 million dollar.

For part b,

To calculate rate, differentiate the given function with respect to x.

$\dfrac{d}{dx}R\left(x\right)=\dfrac{d}{dx}\left (-\left(x\right)^{4}+8\left(x\right)^{3}-38\left(x\right)^{2}+44\left(x\right)\right)$

Applying sum and difference rule of derivative,

$\dfrac{d}{dx}R\left(x\right)=-\dfrac{d}{dx}\left(x^4\right)+\dfrac{d}{dx}\left(8x^3\right)-\dfrac{d}{dx}\left(38x^2\right)+\dfrac{d}{dx}\left(44x\right)$

Applying constant multiple rule of derivative,

$\dfrac{d}{dx}R\left(x\right)=-\dfrac{d}{dx}\left(x^4\right)+8\dfrac{d}{dx}\left(x^3\right)-38\dfrac{d}{dx}\left(x^2\right)+44\dfrac{d}{dx}\left(x\right)$

Applying power rule of derivative,

$\dfrac{d}{dx}R\left(x\right)=-\left(4x^{4-1}\right)+8\left(3x^{3-1}\right)-38\left(2x^{2-1}\right)+44\left(1x^{1-1}\right)$

$\dfrac{d}{dx}R\left(x\right)=-4\left(x^{3}\right)+8\left(3x^{2}\right)-38\left(2x^{1}\right)+44$

$\dfrac{d}{dx}R\left(x\right)=-4x^3+24x^2-76x+44$

Substituting the value x=3,

$\dfrac{d}{dx}R\left(x\right)=-4\left(3\right)^3+24\left(3\right)^2-76\left(3\right)+44$

$\dfrac{d}{dx}R\left(x\right)=-4\left(27\right)+24\left(9\right)-76\left(3\right)+44$

$\dfrac{d}{dx}R\left(x\right)=-108+216-228+44$

$\dfrac{d}{dx}R\left(x\right)=-76$

Negative sign indicates that rate is decreasing.

Rate at which the revenue is changing in the year 2010 is 76 million dollar per year.

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