Answer:
do you have any more information to solve this ?????????
Step-by-step explanation:
Answer:
52.
Step-by-step explanation:
Let x be the number of total pies before the bake sale started.
We have been given that at the beginning of the bake sale, Mary set aside 2 pies for herself.
So the pie left for sale will be x-2.
We are also told that Karen bought half of the remaining pies for her birthday party, so number of pies bought by Karen will be:
.
Further, we are told that 10 more pies were sold during the sale, so number of total pies sold and used by Mary will be:
![2+\frac{x-2}{2}+10](https://tex.z-dn.net/?f=2%2B%5Cfrac%7Bx-2%7D%7B2%7D%2B10)
As there were 15 pies remaining, so we can represent total number of pies in an equation as:
![x=2+\frac{x-2}{2}+10+15](https://tex.z-dn.net/?f=x%3D2%2B%5Cfrac%7Bx-2%7D%7B2%7D%2B10%2B15)
![x=\frac{x-2}{2}+27](https://tex.z-dn.net/?f=x%3D%5Cfrac%7Bx-2%7D%7B2%7D%2B27)
Let us have a common denominator.
![x=\frac{x-2}{2}+\frac{27*2}{2}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7Bx-2%7D%7B2%7D%2B%5Cfrac%7B27%2A2%7D%7B2%7D)
![x=\frac{x-2}{2}+\frac{54}{2}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7Bx-2%7D%7B2%7D%2B%5Cfrac%7B54%7D%7B2%7D)
![x=\frac{x-2+54}{2}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7Bx-2%2B54%7D%7B2%7D)
Let us multiply both sides of our equation by 2.
![2x=\frac{x+52}{2}*2](https://tex.z-dn.net/?f=2x%3D%5Cfrac%7Bx%2B52%7D%7B2%7D%2A2)
![2x=x+52](https://tex.z-dn.net/?f=2x%3Dx%2B52)
![x=52](https://tex.z-dn.net/?f=x%3D52)
Therefore, there were 52 pies before the bake sale started.
Step-by-step explanation:
Should be Emily, Morgan, Kali, Rosa
If you're using the app, try seeing this answer through your browser: brainly.com/question/2867785_______________
Evaluate the indefinite integral:
![\mathsf{\displaystyle\int\! \frac{x}{\sqrt{1-x^4}}\,dx}\\\\\\ \mathsf{=\displaystyle\int\! \frac{1}{2}\cdot 2\cdot \frac{1}{\sqrt{1-(x^2)^2}}\,dx}\\\\\\ \mathsf{=\displaystyle \frac{1}{2}\int\! \frac{1}{\sqrt{1-(x^2)^2}}\cdot 2x\,dx\qquad\quad(i)}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21%20%5Cfrac%7Bx%7D%7B%5Csqrt%7B1-x%5E4%7D%7D%5C%2Cdx%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%3D%5Cdisplaystyle%5Cint%5C%21%20%5Cfrac%7B1%7D%7B2%7D%5Ccdot%202%5Ccdot%20%5Cfrac%7B1%7D%7B%5Csqrt%7B1-%28x%5E2%29%5E2%7D%7D%5C%2Cdx%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%3D%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B2%7D%5Cint%5C%21%20%5Cfrac%7B1%7D%7B%5Csqrt%7B1-%28x%5E2%29%5E2%7D%7D%5Ccdot%202x%5C%2Cdx%5Cqquad%5Cquad%28i%29%7D)
Make a trigonometric substitution:
![\begin{array}{lcl} \mathsf{x^2=sin\,t}&\quad\Rightarrow\quad&\mathsf{2x\,dx=cos\,t\,dt}\\\\ &&\mathsf{t=arcsin(x^2)\,,\qquad 0\ \textless \ x\ \textless \ \frac{\pi}{2}}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Blcl%7D%0A%5Cmathsf%7Bx%5E2%3Dsin%5C%2Ct%7D%26%5Cquad%5CRightarrow%5Cquad%26%5Cmathsf%7B2x%5C%2Cdx%3Dcos%5C%2Ct%5C%2Cdt%7D%5C%5C%5C%5C%0A%26%26%5Cmathsf%7Bt%3Darcsin%28x%5E2%29%5C%2C%2C%5Cqquad%200%5C%20%5Ctextless%20%5C%20x%5C%20%5Ctextless%20%5C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%5Cend%7Barray%7D)
so the integral (i) becomes
![\mathsf{=\displaystyle\frac{1}{2}\int\!\frac{1}{\sqrt{1-sin^2\,t}}\cdot cos\,t\,dt\qquad\quad (but~1-sin^2\,t=cos^2\,t)}\\\\\\ \mathsf{=\displaystyle\frac{1}{2}\int\!\frac{1}{\sqrt{cos^2\,t}}\cdot cos\,t\,dt}](https://tex.z-dn.net/?f=%5Cmathsf%7B%3D%5Cdisplaystyle%5Cfrac%7B1%7D%7B2%7D%5Cint%5C%21%5Cfrac%7B1%7D%7B%5Csqrt%7B1-sin%5E2%5C%2Ct%7D%7D%5Ccdot%20cos%5C%2Ct%5C%2Cdt%5Cqquad%5Cquad%20%28but~1-sin%5E2%5C%2Ct%3Dcos%5E2%5C%2Ct%29%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%3D%5Cdisplaystyle%5Cfrac%7B1%7D%7B2%7D%5Cint%5C%21%5Cfrac%7B1%7D%7B%5Csqrt%7Bcos%5E2%5C%2Ct%7D%7D%5Ccdot%20cos%5C%2Ct%5C%2Cdt%7D)
![\mathsf{=\displaystyle\frac{1}{2}\int\!\frac{1}{cos\,t}\cdot cos\,t\,dt}\\\\\\ \mathsf{=\displaystyle\frac{1}{2}\int\!\f dt}\\\\\\ \mathsf{=\displaystyle\frac{1}{2}\,t+C}](https://tex.z-dn.net/?f=%5Cmathsf%7B%3D%5Cdisplaystyle%5Cfrac%7B1%7D%7B2%7D%5Cint%5C%21%5Cfrac%7B1%7D%7Bcos%5C%2Ct%7D%5Ccdot%20cos%5C%2Ct%5C%2Cdt%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%3D%5Cdisplaystyle%5Cfrac%7B1%7D%7B2%7D%5Cint%5C%21%5Cf%20dt%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7B%3D%5Cdisplaystyle%5Cfrac%7B1%7D%7B2%7D%5C%2Ct%2BC%7D)
Now, substitute back for t = arcsin(x²), and you finally get the result:
![\mathsf{\displaystyle\int\! \frac{x}{\sqrt{1-(x^2)^2}}\,dx=\frac{1}{2}\,arcsin(x^2)+C}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21%20%5Cfrac%7Bx%7D%7B%5Csqrt%7B1-%28x%5E2%29%5E2%7D%7D%5C%2Cdx%3D%5Cfrac%7B1%7D%7B2%7D%5C%2Carcsin%28x%5E2%29%2BC%7D)
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You could also make
x² = cos t
and you would get this expression for the integral:
![\mathsf{\displaystyle\int\! \frac{x}{\sqrt{1-(x^2)^2}}\,dx=-\,\frac{1}{2}\,arccos(x^2)+C_2}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdisplaystyle%5Cint%5C%21%20%5Cfrac%7Bx%7D%7B%5Csqrt%7B1-%28x%5E2%29%5E2%7D%7D%5C%2Cdx%3D-%5C%2C%5Cfrac%7B1%7D%7B2%7D%5C%2Carccos%28x%5E2%29%2BC_2%7D)
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which is fine, because those two functions have the same derivative, as the difference between them is a constant:
![\mathsf{\dfrac{1}{2}\,arcsin(x^2)-\left(-\dfrac{1}{2}\,arccos(x^2)\right)}\\\\\\ =\mathsf{\dfrac{1}{2}\,arcsin(x^2)+\dfrac{1}{2}\,arccos(x^2)}\\\\\\ =\mathsf{\dfrac{1}{2}\cdot \left[\,arcsin(x^2)+arccos(x^2)\right]}\\\\\\ =\mathsf{\dfrac{1}{2}\cdot \dfrac{\pi}{2}}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdfrac%7B1%7D%7B2%7D%5C%2Carcsin%28x%5E2%29-%5Cleft%28-%5Cdfrac%7B1%7D%7B2%7D%5C%2Carccos%28x%5E2%29%5Cright%29%7D%5C%5C%5C%5C%5C%5C%0A%3D%5Cmathsf%7B%5Cdfrac%7B1%7D%7B2%7D%5C%2Carcsin%28x%5E2%29%2B%5Cdfrac%7B1%7D%7B2%7D%5C%2Carccos%28x%5E2%29%7D%5C%5C%5C%5C%5C%5C%0A%3D%5Cmathsf%7B%5Cdfrac%7B1%7D%7B2%7D%5Ccdot%20%5Cleft%5B%5C%2Carcsin%28x%5E2%29%2Barccos%28x%5E2%29%5Cright%5D%7D%5C%5C%5C%5C%5C%5C%0A%3D%5Cmathsf%7B%5Cdfrac%7B1%7D%7B2%7D%5Ccdot%20%5Cdfrac%7B%5Cpi%7D%7B2%7D%7D)
![\mathsf{=\dfrac{\pi}{4}}](https://tex.z-dn.net/?f=%5Cmathsf%7B%3D%5Cdfrac%7B%5Cpi%7D%7B4%7D%7D)
✔
and that constant does not interfer in the differentiation process, because the derivative of a constant is zero.
I hope this helps. =)
The answer is A. If the number of T-shirts is t, then you substitute the t in each equation and solve for all the numbers.
f(t)= 17+13t f(t)= 17+13t f(t)= 17+13t
f(t)= 17+13(12) f(t)= 17+13(17) f(t)= 17+13(22)
f(t)= 17+156 f(t)= 17+221 f(t)= 17+286
f(t)= 173 f(t)= 238 f(t)= 303