your anwser is 3 because its easy on a calculator
Answer:
equal to
Step-by-step explanation:
Proof:
Given any functions
they are linearly dependent if we can find values of
such that
Using the given functions in the above equation we get
![c_{1}f_{1}(x)+c_2f_{2}(x)+c_{3}f_{3}(x)=0\\\\c_{1}x^{2}+c_{2}(1-x^{2})+c_{3}(2+x^{2})=0\\\\\Rightarrow (c_{1}-c_{2}+c_{3})x^{2}+c_{1}+c_{2}+2c_{3}=0](https://tex.z-dn.net/?f=c_%7B1%7Df_%7B1%7D%28x%29%2Bc_2f_%7B2%7D%28x%29%2Bc_%7B3%7Df_%7B3%7D%28x%29%3D0%5C%5C%5C%5Cc_%7B1%7Dx%5E%7B2%7D%2Bc_%7B2%7D%281-x%5E%7B2%7D%29%2Bc_%7B3%7D%282%2Bx%5E%7B2%7D%29%3D0%5C%5C%5C%5C%5CRightarrow%20%28c_%7B1%7D-c_%7B2%7D%2Bc_%7B3%7D%29x%5E%7B2%7D%2Bc_%7B1%7D%2Bc_%7B2%7D%2B2c_%7B3%7D%3D0)
This will be satisfied if and only if
![c_1-c_2+c_3=0,c_1+c_2+2c_3=0](https://tex.z-dn.net/?f=c_1-c_2%2Bc_3%3D0%2Cc_1%2Bc_2%2B2c_3%3D0)
Solving the equations we get
![c_1+2c_3=-c_2\\\\c_1+c_1+2c_3+c_3=0\\2c_1+3c_3=0](https://tex.z-dn.net/?f=c_1%2B2c_3%3D-c_2%5C%5C%5C%5Cc_1%2Bc_1%2B2c_3%2Bc_3%3D0%5C%5C2c_1%2B3c_3%3D0)
Since we have 3 variables and 2 equations thus we will get many solutions
one being if we put
we get
![c_1+2c_3=-c_2\\\\c_1+c_1+2c_3+c_3=0\\2c_1+3c_3=0\\\\c_1=\frac{-3}{2}\\\\c_2=\frac{-1}{2}](https://tex.z-dn.net/?f=c_1%2B2c_3%3D-c_2%5C%5C%5C%5Cc_1%2Bc_1%2B2c_3%2Bc_3%3D0%5C%5C2c_1%2B3c_3%3D0%5C%5C%5C%5Cc_1%3D%5Cfrac%7B-3%7D%7B2%7D%5C%5C%5C%5Cc_2%3D%5Cfrac%7B-1%7D%7B2%7D)
Thus we have
as one solution. Hence the given functions are linearly dependent.
Each lock costs $1.70
The total profit is 244.8
Answer: The force is 6.25 times stronger.
Step-by-step explanation:
The formula to find the force of attraction between any two object is given by :-
(i)
, where M and m are the masses of the objects , d is the distance between them and G is gravitational constant.
If you decrease the distance between two celestial objects by a factor of 2.5, then , the new distance will become
![d'=\dfrac{1}{2.5}d](https://tex.z-dn.net/?f=d%27%3D%5Cdfrac%7B1%7D%7B2.5%7Dd)
Plug this in the above formula , we get
![F'=G\dfrac{Mm}{(\dfrac{1}{2.5}d)^2}\\\\\Rightarrow\ F'=(2.5)^2(G\dfrac{Mm}{d^2})\\\\\Rightarrow\ F'=6.25(G\dfrac{Mm}{d^2})=6.25F----------\text{from }(i)](https://tex.z-dn.net/?f=F%27%3DG%5Cdfrac%7BMm%7D%7B%28%5Cdfrac%7B1%7D%7B2.5%7Dd%29%5E2%7D%5C%5C%5C%5C%5CRightarrow%5C%20F%27%3D%282.5%29%5E2%28G%5Cdfrac%7BMm%7D%7Bd%5E2%7D%29%5C%5C%5C%5C%5CRightarrow%5C%20F%27%3D6.25%28G%5Cdfrac%7BMm%7D%7Bd%5E2%7D%29%3D6.25F----------%5Ctext%7Bfrom%20%7D%28i%29)
Hence, the force is 6.25 times stronger.