The answer is 65
there are 10 millimeters in every centimeter...so to get the size of the grid, you multiply 260 x 10 to get 260 centimeters and then to see how many millimeters are in the 260 centimeter grid you multiply it by 260 by 10, which gives you 2,600 millimeters, so then all that is left to do is divide 2,600 millimeters by 40 millimeters for each of her 40 millimeter equilateral triangles to see how many will fit...the answer is 65 <span />
![y'-\dfrac13y=\dfrac13xe^x\ln x\,y^{-2}](https://tex.z-dn.net/?f=y%27-%5Cdfrac13y%3D%5Cdfrac13xe%5Ex%5Cln%20x%5C%2Cy%5E%7B-2%7D)
Divide both sides by
:
![3y^2y'-y^3=xe^x\ln x](https://tex.z-dn.net/?f=3y%5E2y%27-y%5E3%3Dxe%5Ex%5Cln%20x)
Substitute
, so that
.
![v'-v=xe^x\ln x](https://tex.z-dn.net/?f=v%27-v%3Dxe%5Ex%5Cln%20x)
Multiply both sides by
:
![e^{-x}v'-e^{-x}v=x\ln x](https://tex.z-dn.net/?f=e%5E%7B-x%7Dv%27-e%5E%7B-x%7Dv%3Dx%5Cln%20x)
The left side can be condensed into the derivative of a product.
![(e^{-x}v)'=x\ln x](https://tex.z-dn.net/?f=%28e%5E%7B-x%7Dv%29%27%3Dx%5Cln%20x)
Integrate both sides to get
![e^{-x}v=\dfrac12x^2\ln x-\dfrac14x^2+C](https://tex.z-dn.net/?f=e%5E%7B-x%7Dv%3D%5Cdfrac12x%5E2%5Cln%20x-%5Cdfrac14x%5E2%2BC)
Solve for
:
![v=\dfrac12x^2e^x\ln x-\dfrac14x^2e^x+Ce^x](https://tex.z-dn.net/?f=v%3D%5Cdfrac12x%5E2e%5Ex%5Cln%20x-%5Cdfrac14x%5E2e%5Ex%2BCe%5Ex)
Solve for
:
![y^3=\dfrac12x^2e^x\ln x-\dfrac14x^2e^x+Ce^x](https://tex.z-dn.net/?f=y%5E3%3D%5Cdfrac12x%5E2e%5Ex%5Cln%20x-%5Cdfrac14x%5E2e%5Ex%2BCe%5Ex)
![\implies\boxed{y(x)=\sqrt[3]{\dfrac14x^2e^x(2\ln x-1)+Ce^x}}](https://tex.z-dn.net/?f=%5Cimplies%5Cboxed%7By%28x%29%3D%5Csqrt%5B3%5D%7B%5Cdfrac14x%5E2e%5Ex%282%5Cln%20x-1%29%2BCe%5Ex%7D%7D)
Answer:
Please see attachment
Step-by-step explanation:
Please see attachment