To move something multiplied or divided, do the inverse to both sides. Let’s work with an example: 3x+y=10. Solving for y means that we have to get y by itself. Therefore, we have to move everything else to the other side.
Answer:
The dimensions that minimize the cost of materials for the cylinders have radii of about 3.628 cm and heights of about 7.256 cm.
Step-by-step explanation:
A cylindrical can holds 300 cubic centimeters, and we want to find the dimensions that minimize the cost for materials: that is, the dimensions that minimize the surface area.
Recall that the volume for a cylinder is given by:
![\displaystyle V = \pi r^2h](https://tex.z-dn.net/?f=%5Cdisplaystyle%20V%20%3D%20%5Cpi%20r%5E2h)
Substitute:
![\displaystyle (300) = \pi r^2 h](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%28300%29%20%3D%20%5Cpi%20r%5E2%20h)
Solve for <em>h: </em>
![\displaystyle \frac{300}{\pi r^2} = h](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B300%7D%7B%5Cpi%20r%5E2%7D%20%3D%20h)
Recall that the surface area of a cylinder is given by:
![\displaystyle A = 2\pi r^2 + 2\pi rh](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%202%5Cpi%20r%5E2%20%2B%202%5Cpi%20rh)
We want to minimize this equation. To do so, we can find its critical points, since extrema (minima and maxima) occur at critical points.
First, substitute for <em>h</em>.
![\displaystyle \begin{aligned} A &= 2\pi r^2 + 2\pi r\left(\frac{300}{\pi r^2}\right) \\ \\ &=2\pi r^2 + \frac{600}{ r} \end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%20A%20%26%3D%202%5Cpi%20r%5E2%20%2B%202%5Cpi%20r%5Cleft%28%5Cfrac%7B300%7D%7B%5Cpi%20r%5E2%7D%5Cright%29%20%5C%5C%20%5C%5C%20%26%3D2%5Cpi%20r%5E2%20%2B%20%5Cfrac%7B600%7D%7B%20r%7D%20%20%5Cend%7Baligned%7D)
Find its derivative:
![\displaystyle A' = 4\pi r - \frac{600}{r^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%27%20%3D%204%5Cpi%20r%20-%20%5Cfrac%7B600%7D%7Br%5E2%7D)
Solve for its zero(s):
![\displaystyle \begin{aligned} (0) &= 4\pi r - \frac{600}{r^2} \\ \\ 4\pi r - \frac{600}{r^2} &= 0 \\ \\ 4\pi r^3 - 600 &= 0 \\ \\ \pi r^3 &= 150 \\ \\ r &= \sqrt[3]{\frac{150}{\pi}} \approx 3.628\text{ cm}\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%20%280%29%20%26%3D%204%5Cpi%20r%20%20-%20%5Cfrac%7B600%7D%7Br%5E2%7D%20%5C%5C%20%5C%5C%204%5Cpi%20r%20-%20%5Cfrac%7B600%7D%7Br%5E2%7D%20%26%3D%200%20%5C%5C%20%5C%5C%204%5Cpi%20r%5E3%20-%20600%20%26%3D%200%20%5C%5C%20%5C%5C%20%5Cpi%20r%5E3%20%26%3D%20150%20%5C%5C%20%5C%5C%20r%20%26%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7B150%7D%7B%5Cpi%7D%7D%20%5Capprox%203.628%5Ctext%7B%20cm%7D%5Cend%7Baligned%7D)
Hence, the radius that minimizes the surface area will be about 3.628 centimeters.
Then the height will be:
![\displaystyle \begin{aligned} h&= \frac{300}{\pi\left( \sqrt[3]{\dfrac{150}{\pi}}\right)^2} \\ \\ &= \frac{60}{\pi \sqrt[3]{\dfrac{180}{\pi^2}}}\approx 7.25 6\text{ cm} \end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%20%5Cbegin%7Baligned%7D%20h%26%3D%20%5Cfrac%7B300%7D%7B%5Cpi%5Cleft%28%20%5Csqrt%5B3%5D%7B%5Cdfrac%7B150%7D%7B%5Cpi%7D%7D%5Cright%29%5E2%7D%20%20%5C%5C%20%5C%5C%20%26%3D%20%5Cfrac%7B60%7D%7B%5Cpi%20%5Csqrt%5B3%5D%7B%5Cdfrac%7B180%7D%7B%5Cpi%5E2%7D%7D%7D%5Capprox%207.25%206%5Ctext%7B%20cm%7D%20%20%20%5Cend%7Baligned%7D)
In conclusion, the dimensions that minimize the cost of materials for the cylinders have radii of about 3.628 cm and heights of about 7.256 cm.
Hannah would need 4 boxes
8x12 to find out how many colored pencils in each box you would get 96 then divided that by 350 to see how many boxes you would need you would get 3.645... and so on you would need to round up since Hannah needs 350 she can’t have less but she can have more therefore Hannah needs 4 boxes of pencils
Answer:
(44, - 27 )
Step-by-step explanation:
Given the 2 equations
5x + 8y = 4 → (1)
2x + 3y = 7 → (2)
Multiplying (1) by 2 and (2) by - 5 , then adding the result will eliminate x
10x + 16y = 8 → (3)
- 10x - 15y = - 35 → (4)
Add (3) and (4) term by term to eliminate x
y = - 27
Substitute y = - 27 into either of the 2 equations and solve for x
Substituting into (2)
2x + 3(- 27) = 7
2x - 81 = 7 ( add 81 to both sides )
2x = 88 ( divide both sides by 2 )
x = 44
solution is (44, - 27 )
Answer:
50.27
Step-by-step explanation:
perimeter= length of arc + 2r= 14.28
length of arc= rx where x is the angle which is pi/2
radius =4
area= pi* r^2= 50.27