Answer:
21
Step-by-step explanation:
Answer:
The maximum volume of the box is:
![V =\frac{5}{3}\sqrt{\frac{5}{3}}](https://tex.z-dn.net/?f=V%20%3D%5Cfrac%7B5%7D%7B3%7D%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D)
Step-by-step explanation:
Given
![Surface\ Area = 10m^2](https://tex.z-dn.net/?f=Surface%5C%20Area%20%3D%2010m%5E2)
Required
The maximum volume of the box
Let
![a \to base\ dimension](https://tex.z-dn.net/?f=a%20%5Cto%20base%5C%20dimension)
![b \to height](https://tex.z-dn.net/?f=b%20%5Cto%20height)
The surface area of the box is:
![Surface\ Area = 2(a*a + a*b + a*b)](https://tex.z-dn.net/?f=Surface%5C%20Area%20%3D%202%28a%2Aa%20%2B%20a%2Ab%20%2B%20a%2Ab%29)
![Surface\ Area = 2(a^2 + ab + ab)](https://tex.z-dn.net/?f=Surface%5C%20Area%20%3D%202%28a%5E2%20%2B%20ab%20%2B%20ab%29)
![Surface\ Area = 2(a^2 + 2ab)](https://tex.z-dn.net/?f=Surface%5C%20Area%20%3D%202%28a%5E2%20%2B%202ab%29)
So, we have:
![2(a^2 + 2ab) = 10](https://tex.z-dn.net/?f=2%28a%5E2%20%2B%202ab%29%20%3D%2010)
Divide both sides by 2
![a^2 + 2ab = 5](https://tex.z-dn.net/?f=a%5E2%20%2B%202ab%20%3D%205)
Make b the subject
![2ab = 5 -a^2](https://tex.z-dn.net/?f=2ab%20%3D%205%20-a%5E2)
![b = \frac{5 -a^2}{2a}](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B5%20-a%5E2%7D%7B2a%7D)
The volume of the box is:
![V = a*a*b](https://tex.z-dn.net/?f=V%20%3D%20a%2Aa%2Ab)
![V = a^2b](https://tex.z-dn.net/?f=V%20%3D%20a%5E2b)
Substitute: ![b = \frac{5 -a^2}{2a}](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B5%20-a%5E2%7D%7B2a%7D)
![V = a^2*\frac{5 - a^2}{2a}](https://tex.z-dn.net/?f=V%20%3D%20a%5E2%2A%5Cfrac%7B5%20-%20a%5E2%7D%7B2a%7D)
![V = a*\frac{5 - a^2}{2}](https://tex.z-dn.net/?f=V%20%3D%20a%2A%5Cfrac%7B5%20-%20a%5E2%7D%7B2%7D)
![V = \frac{5a - a^3}{2}](https://tex.z-dn.net/?f=V%20%3D%20%5Cfrac%7B5a%20-%20a%5E3%7D%7B2%7D)
Spit
![V = \frac{5a}{2} - \frac{a^3}{2}](https://tex.z-dn.net/?f=V%20%3D%20%5Cfrac%7B5a%7D%7B2%7D%20-%20%5Cfrac%7Ba%5E3%7D%7B2%7D)
Differentiate V with respect to a
![V' = \frac{5}{2} -3 * \frac{a^2}{2}](https://tex.z-dn.net/?f=V%27%20%3D%20%5Cfrac%7B5%7D%7B2%7D%20-3%20%2A%20%5Cfrac%7Ba%5E2%7D%7B2%7D)
![V' = \frac{5}{2} -\frac{3a^2}{2}](https://tex.z-dn.net/?f=V%27%20%3D%20%5Cfrac%7B5%7D%7B2%7D%20-%5Cfrac%7B3a%5E2%7D%7B2%7D)
Set
to calculate a
![0 = \frac{5}{2} -\frac{3a^2}{2}](https://tex.z-dn.net/?f=0%20%3D%20%5Cfrac%7B5%7D%7B2%7D%20-%5Cfrac%7B3a%5E2%7D%7B2%7D)
Collect like terms
![\frac{3a^2}{2} = \frac{5}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B3a%5E2%7D%7B2%7D%20%3D%20%5Cfrac%7B5%7D%7B2%7D)
Multiply both sides by 2
![3a^2= 5](https://tex.z-dn.net/?f=3a%5E2%3D%205)
Solve for a
![a^2= \frac{5}{3}](https://tex.z-dn.net/?f=a%5E2%3D%20%5Cfrac%7B5%7D%7B3%7D)
![a= \sqrt{\frac{5}{3}}](https://tex.z-dn.net/?f=a%3D%20%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D)
Recall that:
![b = \frac{5 -a^2}{2a}](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B5%20-a%5E2%7D%7B2a%7D)
![b = \frac{5 -(\sqrt{\frac{5}{3}})^2}{2*\sqrt{\frac{5}{3}}}](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B5%20-%28%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D%29%5E2%7D%7B2%2A%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D%7D)
![b = \frac{5 -\frac{5}{3}}{2*\sqrt{\frac{5}{3}}}](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B5%20-%5Cfrac%7B5%7D%7B3%7D%7D%7B2%2A%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D%7D)
![b = \frac{\frac{15 - 5}{3}}{2*\sqrt{\frac{5}{3}}}](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B%5Cfrac%7B15%20-%205%7D%7B3%7D%7D%7B2%2A%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D%7D)
![b = \frac{\frac{10}{3}}{2*\sqrt{\frac{5}{3}}}](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B%5Cfrac%7B10%7D%7B3%7D%7D%7B2%2A%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D%7D)
![b = \frac{\frac{5}{3}}{\sqrt{\frac{5}{3}}}](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B%5Cfrac%7B5%7D%7B3%7D%7D%7B%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D%7D)
Apply law of indices
![b = (\frac{5}{3})^{1 - \frac{1}{2}}](https://tex.z-dn.net/?f=b%20%3D%20%28%5Cfrac%7B5%7D%7B3%7D%29%5E%7B1%20-%20%5Cfrac%7B1%7D%7B2%7D%7D)
![b = (\frac{5}{3})^{\frac{1}{2}}](https://tex.z-dn.net/?f=b%20%3D%20%28%5Cfrac%7B5%7D%7B3%7D%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D)
![b = \sqrt{\frac{5}{3}}](https://tex.z-dn.net/?f=b%20%3D%20%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D)
So:
![V = a^2b](https://tex.z-dn.net/?f=V%20%3D%20a%5E2b)
![V =\sqrt{(\frac{5}{3})^2} * \sqrt{\frac{5}{3}}](https://tex.z-dn.net/?f=V%20%3D%5Csqrt%7B%28%5Cfrac%7B5%7D%7B3%7D%29%5E2%7D%20%2A%20%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D)
![V =\frac{5}{3} * \sqrt{\frac{5}{3}}](https://tex.z-dn.net/?f=V%20%3D%5Cfrac%7B5%7D%7B3%7D%20%2A%20%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D)
![V =\frac{5}{3}\sqrt{\frac{5}{3}}](https://tex.z-dn.net/?f=V%20%3D%5Cfrac%7B5%7D%7B3%7D%5Csqrt%7B%5Cfrac%7B5%7D%7B3%7D%7D)
to get the equation of a line, we simply need two points, say for the Red one ... notice in the graph the lines passes through (0,2) and (-1,6), so let's use those
![\bf (\stackrel{x_1}{0}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{-1}~,~\stackrel{y_2}{6}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{6-2}{-1-0}\implies \cfrac{4}{-1}\implies -4 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-2=-4(x-0) \\\\\\ y-2=-4x\implies \blacktriangleright y=-4x+2 \blacktriangleleft](https://tex.z-dn.net/?f=%5Cbf%20%28%5Cstackrel%7Bx_1%7D%7B0%7D~%2C~%5Cstackrel%7By_1%7D%7B2%7D%29%5Cqquad%20%28%5Cstackrel%7Bx_2%7D%7B-1%7D~%2C~%5Cstackrel%7By_2%7D%7B6%7D%29%20%5C%5C%5C%5C%5C%5C%20slope%20%3D%20m%5Cimplies%20%5Ccfrac%7B%5Cstackrel%7Brise%7D%7B%20y_2-%20y_1%7D%7D%7B%5Cstackrel%7Brun%7D%7B%20x_2-%20x_1%7D%7D%5Cimplies%20%5Ccfrac%7B6-2%7D%7B-1-0%7D%5Cimplies%20%5Ccfrac%7B4%7D%7B-1%7D%5Cimplies%20-4%20%5C%5C%5C%5C%5C%5C%20%5Cbegin%7Barray%7D%7B%7Cc%7Cll%7D%20%5Ccline%7B1-1%7D%20%5Ctextit%7Bpoint-slope%20form%7D%5C%5C%20%5Ccline%7B1-1%7D%20%5C%5C%20y-y_1%3Dm%28x-x_1%29%20%5C%5C%5C%5C%20%5Ccline%7B1-1%7D%20%5Cend%7Barray%7D%5Cimplies%20y-2%3D-4%28x-0%29%20%5C%5C%5C%5C%5C%5C%20y-2%3D-4x%5Cimplies%20%5Cblacktriangleright%20y%3D-4x%2B2%20%5Cblacktriangleleft)
now, for the Blue one, say let's use hmmm it passes through (0,2) and (1.6)
![\bf (\stackrel{x_1}{0}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{1}~,~\stackrel{y_2}{6}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{6-2}{1-0}\implies \cfrac{4}{1}\implies 4 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-2=4(x-0) \\\\\\ y-2=4x\implies \blacktriangleright y=4x+2 \blacktriangleleft](https://tex.z-dn.net/?f=%5Cbf%20%28%5Cstackrel%7Bx_1%7D%7B0%7D~%2C~%5Cstackrel%7By_1%7D%7B2%7D%29%5Cqquad%20%28%5Cstackrel%7Bx_2%7D%7B1%7D~%2C~%5Cstackrel%7By_2%7D%7B6%7D%29%20%5C%5C%5C%5C%5C%5C%20slope%20%3D%20m%5Cimplies%20%5Ccfrac%7B%5Cstackrel%7Brise%7D%7B%20y_2-%20y_1%7D%7D%7B%5Cstackrel%7Brun%7D%7B%20x_2-%20x_1%7D%7D%5Cimplies%20%5Ccfrac%7B6-2%7D%7B1-0%7D%5Cimplies%20%5Ccfrac%7B4%7D%7B1%7D%5Cimplies%204%20%5C%5C%5C%5C%5C%5C%20%5Cbegin%7Barray%7D%7B%7Cc%7Cll%7D%20%5Ccline%7B1-1%7D%20%5Ctextit%7Bpoint-slope%20form%7D%5C%5C%20%5Ccline%7B1-1%7D%20%5C%5C%20y-y_1%3Dm%28x-x_1%29%20%5C%5C%5C%5C%20%5Ccline%7B1-1%7D%20%5Cend%7Barray%7D%5Cimplies%20y-2%3D4%28x-0%29%20%5C%5C%5C%5C%5C%5C%20y-2%3D4x%5Cimplies%20%5Cblacktriangleright%20y%3D4x%2B2%20%5Cblacktriangleleft)
Well to find the answer add 23+17+29+28.
Your total/answer will be 97 pounds of food.
Answer:
6, 9, 15, 24, 36, 51, 69, 90, 114