Answer:
- 1/4
Step-by-step explanation:
Answer:
![\large\boxed{1.64=1\dfrac{16}{25}}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7B1.64%3D1%5Cdfrac%7B16%7D%7B25%7D%7D)
Step-by-step explanation:
![1.64=1+0.\underbrace{64}_2=1+\dfrac{64}{1\underbrace{00}_2}=1\dfrac{64}{100}=1\dfrac{64:4}{100:4}=1\dfrac{16}{25}](https://tex.z-dn.net/?f=1.64%3D1%2B0.%5Cunderbrace%7B64%7D_2%3D1%2B%5Cdfrac%7B64%7D%7B1%5Cunderbrace%7B00%7D_2%7D%3D1%5Cdfrac%7B64%7D%7B100%7D%3D1%5Cdfrac%7B64%3A4%7D%7B100%3A4%7D%3D1%5Cdfrac%7B16%7D%7B25%7D)
Answer:
a)12 b)12 c)36
Step-by-step explanation:
a)find the least common multiple. the number of marbles is that number * n (where n is an integer)
b) the least common multiple. it is 4*3=...
c)12*3 =?
then just divide by 2,3,4 to get the answer
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dy
Find —— for an implicit function:
dx
x²y – 3x = y³ – 3
First, differentiate implicitly both sides with respect to x. Keep in mind that y is not just a variable, but it is also a function of x, so you have to use the chain rule there:
![\mathsf{\dfrac{d}{dx}(x^2 y-3x)=\dfrac{d}{dx}(y^3-3)}\\\\\\ \mathsf{\dfrac{d}{dx}(x^2 y)-3\,\dfrac{d}{dx}(x)=\dfrac{d}{dx}(y^3)-\dfrac{d}{dx}(3)}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdfrac%7Bd%7D%7Bdx%7D%28x%5E2%20y-3x%29%3D%5Cdfrac%7Bd%7D%7Bdx%7D%28y%5E3-3%29%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%5Cdfrac%7Bd%7D%7Bdx%7D%28x%5E2%20y%29-3%5C%2C%5Cdfrac%7Bd%7D%7Bdx%7D%28x%29%3D%5Cdfrac%7Bd%7D%7Bdx%7D%28y%5E3%29-%5Cdfrac%7Bd%7D%7Bdx%7D%283%29%7D)
Applying the product rule for the first term at the left-hand side:
![\mathsf{\left[\dfrac{d}{dx}(x^2)\cdot y+x^2\cdot \dfrac{d}{dx}(y)\right]-3\cdot 1=3y^2\cdot \dfrac{dy}{dx}-0}\\\\\\ \mathsf{\left[2x\cdot y+x^2\cdot \dfrac{dy}{dx}\right]-3=3y^2\cdot \dfrac{dy}{dx}}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cleft%5B%5Cdfrac%7Bd%7D%7Bdx%7D%28x%5E2%29%5Ccdot%20y%2Bx%5E2%5Ccdot%20%5Cdfrac%7Bd%7D%7Bdx%7D%28y%29%5Cright%5D-3%5Ccdot%201%3D3y%5E2%5Ccdot%20%5Cdfrac%7Bdy%7D%7Bdx%7D-0%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%5Cleft%5B2x%5Ccdot%20y%2Bx%5E2%5Ccdot%20%5Cdfrac%7Bdy%7D%7Bdx%7D%5Cright%5D-3%3D3y%5E2%5Ccdot%20%5Cdfrac%7Bdy%7D%7Bdx%7D%7D)
dy
Now, isolate —— in the equation above:
dx
![\mathsf{2xy+x^2\cdot \dfrac{dy}{dx}-3=3y^2\cdot \dfrac{dy}{dx}}\\\\\\ \mathsf{2xy+x^2\cdot \dfrac{dy}{dx}-3-3y^2\cdot \dfrac{dy}{dx}=0}\\\\\\ \mathsf{x^2\cdot \dfrac{dy}{dx}-3y^2\cdot \dfrac{dy}{dx}=-\,2xy+3}\\\\\\ \mathsf{(x^2-3y^2)\cdot \dfrac{dy}{dx}=-\,2xy+3}](https://tex.z-dn.net/?f=%5Cmathsf%7B2xy%2Bx%5E2%5Ccdot%20%5Cdfrac%7Bdy%7D%7Bdx%7D-3%3D3y%5E2%5Ccdot%20%5Cdfrac%7Bdy%7D%7Bdx%7D%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B2xy%2Bx%5E2%5Ccdot%20%5Cdfrac%7Bdy%7D%7Bdx%7D-3-3y%5E2%5Ccdot%20%5Cdfrac%7Bdy%7D%7Bdx%7D%3D0%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7Bx%5E2%5Ccdot%20%5Cdfrac%7Bdy%7D%7Bdx%7D-3y%5E2%5Ccdot%20%5Cdfrac%7Bdy%7D%7Bdx%7D%3D-%5C%2C2xy%2B3%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%28x%5E2-3y%5E2%29%5Ccdot%20%5Cdfrac%7Bdy%7D%7Bdx%7D%3D-%5C%2C2xy%2B3%7D)
![\mathsf{\dfrac{dy}{dx}=\dfrac{-\,2xy+3}{x^2-3y^2}\qquad\quad for~~x^2-3y^2\ne 0}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdfrac%7Bdy%7D%7Bdx%7D%3D%5Cdfrac%7B-%5C%2C2xy%2B3%7D%7Bx%5E2-3y%5E2%7D%5Cqquad%5Cquad%20for~~x%5E2-3y%5E2%5Cne%200%7D)
Compute the derivative value at the point (– 1, 2):
x = – 1 and y = 2
![\mathsf{\left.\dfrac{dy}{dx}\right|_{(-1,\,2)}=\dfrac{-\,2\cdot (-1)\cdot 2+3}{(-1)^2-3\cdot 2^2}}\\\\\\ \mathsf{\left.\dfrac{dy}{dx}\right|_{(-1,\,2)}=\dfrac{4+3}{1-12}}\\\\\\ \mathsf{\left.\dfrac{dy}{dx}\right|_{(-1,\,2)}=\dfrac{7}{-11}}\\\\\\\\ \therefore~~\mathsf{\left.\dfrac{dy}{dx}\right|_{(-1,\,2)}=-\,\dfrac{7}{11}}\quad\longleftarrow\quad\textsf{this is the answer.}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cleft.%5Cdfrac%7Bdy%7D%7Bdx%7D%5Cright%7C_%7B%28-1%2C%5C%2C2%29%7D%3D%5Cdfrac%7B-%5C%2C2%5Ccdot%20%28-1%29%5Ccdot%202%2B3%7D%7B%28-1%29%5E2-3%5Ccdot%202%5E2%7D%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%5Cleft.%5Cdfrac%7Bdy%7D%7Bdx%7D%5Cright%7C_%7B%28-1%2C%5C%2C2%29%7D%3D%5Cdfrac%7B4%2B3%7D%7B1-12%7D%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%5Cleft.%5Cdfrac%7Bdy%7D%7Bdx%7D%5Cright%7C_%7B%28-1%2C%5C%2C2%29%7D%3D%5Cdfrac%7B7%7D%7B-11%7D%7D%5C%5C%5C%5C%5C%5C%5C%5C%20%5Ctherefore~~%5Cmathsf%7B%5Cleft.%5Cdfrac%7Bdy%7D%7Bdx%7D%5Cright%7C_%7B%28-1%2C%5C%2C2%29%7D%3D-%5C%2C%5Cdfrac%7B7%7D%7B11%7D%7D%5Cquad%5Clongleftarrow%5Cquad%5Ctextsf%7Bthis%20is%20the%20answer.%7D)
I hope this helps. =)
Tags: <em>implicit function derivative implicit differentiation chain product rule differential integral calculus</em>
Answer:
The unit rate is:
Step-by-step explanation:
As Chet was eating 1/4 of a turkey every 2 hours.
This tells us that every two hours, Chet will eat 175 1/4 of a turkey.
- Unit rate basically means 'how much of something per 1 unit of something else'.
A unit rate is a rate with a denominator of 1 unit.
This means, in 1 hour, Chet will eat half of what he eats every 2 hours.
so Chet will eat
of a turkey in 1 hour.
i.e.
Therefore, the unit rate is: