In the last census made in a farm, the number of chickens was determined as 8600 and the number of turkeys as 6400. In this farm, the number of chickens decreases by 120 every month, while the number of turkeys increases by 80. Accordingly, after how many months after the last census on this farm will the number of chickens equal the number of turkeys?
A) 7
B) 8
C) 9
D) 10
E) 11
Let $m$ be the number of months that have passed since the last census. Then the number of chickens is $8600-120m$ and the number of turkeys is $6400+80m$. We want to find when $8600-120m=6400+80m$. Simplifying, we get $200m=2200$, so $m=11$. Therefore, the answer is $\boxed{\textbf{(E)}\ 11}$.
A hyperbola centred at (0,0) has verices (0,+-6) and one focus (0,-10) what is the standard form equation of the hyperbola.
The distance between the vertices is $2a=12$, where $a$ is the distance from the origin to either vertex. Since the center is the origin, $a = 6$. The distance from the center to a focus is $c=10$. The distance between a focus and vertex is $ae=10-6\implies e = 1.$ Thus we have the relation $b^2 = c^2-a^2 = 64.$ Therefore, the equation of the hyperbola is \[\dfrac{x^2}{36} - \dfrac{y^2}{64} = 1.\]
Let's calculate the number of months it would take for the number of chickens to equal the number of turkeys.
Initially, there are 8600 chickens and 6400 turkeys.
In each month, the number of chickens decreases by 120 and the number of turkeys increases by 80.
So, after n months, the number of chickens would be:
8600 - 120n
And the number of turkeys would be:
6400 + 80n
We want to find the number of months when these two values are equal:
8600 - 120n = 6400 + 80n
To solve this equation, let's first simplify:
2200 = 200n
Now, divide both sides by 200:
11 = n
Therefore, after 11 months, the number of chickens will equal the number of turkeys.
Therefore, the answer is:
E) 11