10. A 11. D 12. A (wasn’t too sure about question 12)
Answer:
468.74.
Step-by-step explanation:
We have been given that z varies directly as x and inversely as y. We can represent this information in an equation as:
![z=\frac{kx}{y}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bkx%7D%7By%7D)
First of all, we will find constant of variation using our given equation.
![226=\frac{k\cdot 3}{8}](https://tex.z-dn.net/?f=226%3D%5Cfrac%7Bk%5Ccdot%203%7D%7B8%7D)
![226\cdot \frac{8}{3}=\frac{k\cdot 3}{8}\cdot \frac{8}{3}](https://tex.z-dn.net/?f=226%5Ccdot%20%5Cfrac%7B8%7D%7B3%7D%3D%5Cfrac%7Bk%5Ccdot%203%7D%7B8%7D%5Ccdot%20%5Cfrac%7B8%7D%7B3%7D)
![\frac{1808}{3}=k](https://tex.z-dn.net/?f=%5Cfrac%7B1808%7D%7B3%7D%3Dk)
Upon substituting our given values in our equation, we will get:
![z=\frac{\frac{1808}{3}\cdot 7}{9}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7B%5Cfrac%7B1808%7D%7B3%7D%5Ccdot%207%7D%7B9%7D)
![z=\frac{1808\cdot 7}{9\cdot 3}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7B1808%5Ccdot%207%7D%7B9%5Ccdot%203%7D)
![z=\frac{12656}{27}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7B12656%7D%7B27%7D)
![z=468.740\approx 468.74](https://tex.z-dn.net/?f=z%3D468.740%5Capprox%20468.74)
Therefore, z is approximately 468.74.
The maximum volume of the box is 40√(10/27) cu in.
Here we see that volume is to be maximized
The surface area of the box is 40 sq in
Since the top lid is open, the surface area will be
lb + 2lh + 2bh = 40
Now, the length is equal to the breadth.
Let them be x in
Hence,
x² + 2xh + 2xh = 40
or, 4xh = 40 - x²
or, h = 10/x - x/4
Let f(x) = volume of the box
= lbh
Hence,
f(x) = x²(10/x - x/4)
= 10x - x³/4
differentiating with respect to x and equating it to 0 gives us
f'(x) = 10 - 3x²/4 = 0
or, 3x²/4 = 10
or, x² = 40/3
Hence x will be equal to 2√(10/3)
Now to check whether this value of x will give us the max volume, we will find
f"(2√(10/3))
f"(x) = -3x/2
hence,
f"(2√(10/3)) = -3√(10/3)
Since the above value is negative, volume is maximum for x = 2√(10/3)
Hence volume
= 10 X 2√(10/3) - [2√(10/3)]³/4
= 2√(10/3) [10 - 10/3]
= 2√(10/3) X 20/3
= 40√(10/27) cu in
To learn more about Maximization visit
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Complete Question
(Image Attached)
The answer is x , y = (2, 0)