Answer:
Step-by-step explanation:
Just divide the larger mass by the smaller.
0.5 / 0.005 = 0.5 / 0.5 * 10^-2 = 1 * 10^2 = 100
It take 100 ants to equal the mass of a grasshopper.
Answer:
The partial fraction decomposition is
.
Step-by-step explanation:
Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of "decomposing" the final expression into its initial polynomial fractions.
To find the partial fraction decomposition of
:
First, the form of the partial fraction decomposition is
![\frac{- 4 x^{2} + 13 x - 12}{\left(x + 1\right)^{2} \left(x + 2\right)}=\frac{A}{x + 1}+\frac{B}{\left(x + 1\right)^{2}}+\frac{C}{x + 2}](https://tex.z-dn.net/?f=%5Cfrac%7B-%204%20x%5E%7B2%7D%20%2B%2013%20x%20-%2012%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20%5Cleft%28x%20%2B%202%5Cright%29%7D%3D%5Cfrac%7BA%7D%7Bx%20%2B%201%7D%2B%5Cfrac%7BB%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%7D%2B%5Cfrac%7BC%7D%7Bx%20%2B%202%7D)
Write the right-hand side as a single fraction:
![\frac{- 4 x^{2} + 13 x - 12}{\left(x + 1\right)^{2} \left(x + 2\right)}=\frac{\left(x + 1\right)^{2} C + \left(x + 1\right) \left(x + 2\right) A + \left(x + 2\right) B}{\left(x + 1\right)^{2} \left(x + 2\right)}](https://tex.z-dn.net/?f=%5Cfrac%7B-%204%20x%5E%7B2%7D%20%2B%2013%20x%20-%2012%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20%5Cleft%28x%20%2B%202%5Cright%29%7D%3D%5Cfrac%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20C%20%2B%20%5Cleft%28x%20%2B%201%5Cright%29%20%5Cleft%28x%20%2B%202%5Cright%29%20A%20%2B%20%5Cleft%28x%20%2B%202%5Cright%29%20B%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20%5Cleft%28x%20%2B%202%5Cright%29%7D)
The denominators are equal, so we require the equality of the numerators:
![- 4 x^{2} + 13 x - 12=\left(x + 1\right)^{2} C + \left(x + 1\right) \left(x + 2\right) A + \left(x + 2\right) B](https://tex.z-dn.net/?f=-%204%20x%5E%7B2%7D%20%2B%2013%20x%20-%2012%3D%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20C%20%2B%20%5Cleft%28x%20%2B%201%5Cright%29%20%5Cleft%28x%20%2B%202%5Cright%29%20A%20%2B%20%5Cleft%28x%20%2B%202%5Cright%29%20B)
Expand the right-hand side:
![- 4 x^{2} + 13 x - 12=x^{2} A + x^{2} C + 3 x A + x B + 2 x C + 2 A + 2 B + C](https://tex.z-dn.net/?f=-%204%20x%5E%7B2%7D%20%2B%2013%20x%20-%2012%3Dx%5E%7B2%7D%20A%20%2B%20x%5E%7B2%7D%20C%20%2B%203%20x%20A%20%2B%20x%20B%20%2B%202%20x%20C%20%2B%202%20A%20%2B%202%20B%20%2B%20C)
The coefficients near the like terms should be equal, so the following system is obtained:
![\begin{cases} A + C = -4\\3 A + B + 2 C = 13\\2 A + 2 B + C = -12 \end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%20A%20%2B%20C%20%3D%20-4%5C%5C3%20A%20%2B%20B%20%2B%202%20C%20%3D%2013%5C%5C2%20A%20%2B%202%20B%20%2B%20C%20%3D%20-12%20%5Cend%7Bcases%7D)
Solving this system, we get that
.
Therefore,
![\frac{- 4 x^{2} + 13 x - 12}{\left(x + 1\right)^{2} \left(x + 2\right)}=\frac{50}{x + 1}+\frac{-29}{\left(x + 1\right)^{2}}+\frac{-54}{x + 2}](https://tex.z-dn.net/?f=%5Cfrac%7B-%204%20x%5E%7B2%7D%20%2B%2013%20x%20-%2012%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20%5Cleft%28x%20%2B%202%5Cright%29%7D%3D%5Cfrac%7B50%7D%7Bx%20%2B%201%7D%2B%5Cfrac%7B-29%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%7D%2B%5Cfrac%7B-54%7D%7Bx%20%2B%202%7D)
Answer:
56
Step-by-step explanation:
T+B = 119
T=B-7
B-7 + B = 119
2B = 126
B = 63 years old
T= 63-7 = 56 years old
.75x-14=-8
Use inverse operations to find x
Add 14 on both sides
.75x=6
divide by .75 on both sides
x=8
Answer:
13
Step-by-step explanation: