Given:
The diameter of the right cylinder is 2x cm.
The total surface area is 96 cm cube.
The radius is calculated as,
The total surface area is,
Volume is,
Now, differentiate with respect to x,
Now,
![\begin{gathered} \frac{dV}{dx}=0 \\ 84-3\pi(x^2)=0 \\ x^2=\frac{16}{\pi} \\ x=\sqrt[]{\frac{16}{\pi}} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cfrac%7BdV%7D%7Bdx%7D%3D0%20%5C%5C%2084-3%5Cpi%28x%5E2%29%3D0%20%5C%5C%20x%5E2%3D%5Cfrac%7B16%7D%7B%5Cpi%7D%20%5C%5C%20x%3D%5Csqrt%5B%5D%7B%5Cfrac%7B16%7D%7B%5Cpi%7D%7D%20%5Cend%7Bgathered%7D)
Now, differentiate (1) with respect to x again,
![\begin{gathered} \frac{d^2V}{dx^2}=\frac{d}{dx}(84-3\pi(x^2)) \\ =-6\pi x \\ At\text{ x=}\sqrt[]{\frac{16}{\pi}} \\ \frac{d^2V}{dx^2}=-6\pi\sqrt[]{\frac{16}{\pi}}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cfrac%7Bd%5E2V%7D%7Bdx%5E2%7D%3D%5Cfrac%7Bd%7D%7Bdx%7D%2884-3%5Cpi%28x%5E2%29%29%20%5C%5C%20%3D-6%5Cpi%20x%20%5C%5C%20At%5Ctext%7B%20x%3D%7D%5Csqrt%5B%5D%7B%5Cfrac%7B16%7D%7B%5Cpi%7D%7D%20%5C%5C%20%5Cfrac%7Bd%5E2V%7D%7Bdx%5E2%7D%3D-6%5Cpi%5Csqrt%5B%5D%7B%5Cfrac%7B16%7D%7B%5Cpi%7D%7D%3C0%20%5C%5C%20%20%5Cend%7Bgathered%7D)
Since, the double derivative is negative.
![So,\text{ the volume is maximum at }\sqrt[]{\frac{16}{\pi}}](https://tex.z-dn.net/?f=So%2C%5Ctext%7B%20the%20volume%20is%20maximum%20at%20%7D%5Csqrt%5B%5D%7B%5Cfrac%7B16%7D%7B%5Cpi%7D%7D)
So, the volume becomes,
![\begin{gathered} V=\pi(x^2)h \\ V=\pi(\sqrt[]{\frac{16}{\pi}})^2h \\ V=\frac{16h}{\pi} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20V%3D%5Cpi%28x%5E2%29h%20%5C%5C%20V%3D%5Cpi%28%5Csqrt%5B%5D%7B%5Cfrac%7B16%7D%7B%5Cpi%7D%7D%29%5E2h%20%5C%5C%20V%3D%5Cfrac%7B16h%7D%7B%5Cpi%7D%20%5Cend%7Bgathered%7D)
Answer: maximum volume of the cylinder is,
Answer:
D.120
Step-by-step explanation:
The sides AB and AC are equal and thus the angles they form with BC should be equal . Since now angle ABC is 30 then application of sum of angles in a triangle as a constant(180 degrees) we can subtract 60 degrees from it which gives 120
Answer:
-2.5
Step-by-step explanation:
The point is between -2 & -3
If the lowest temp. is 60 degrees the highest temp i would think would be i don't know maybe 80 or 70 degrees i don't know though might be wrong sorry if you get it wrong
Answer:
Step-by-step explanation:
Our friend asking what the actual function is has a point. I completed this under the assumption that what we have is:
and used the quotient rule to find the derivative, as follows:
![f'(x)=\frac{e^{3x}(5)-[(5x-2)(3e^{3x})]}{(5x-2)^2}](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7Be%5E%7B3x%7D%285%29-%5B%285x-2%29%283e%5E%7B3x%7D%29%5D%7D%7B%285x-2%29%5E2%7D)
and simplifying a bit:
![f'(x)=\frac{5e^{3x}-[15xe^{3x}-6e^{3x}]}{(5x-2)^2}](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7B5e%5E%7B3x%7D-%5B15xe%5E%7B3x%7D-6e%5E%7B3x%7D%5D%7D%7B%285x-2%29%5E2%7D)
and a bit more to:
and combining like terms:
and factor out the GFC in the numerator to get:
That's the derivative simplified. If we want f'(0), we sub in 0's for the x's in there and get the value of the derivative at x = 0:
which simplifies to
which translates to
The slope of the function is 11/4 at the point (0, -1/2)
