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: the nots confused me y’ is that integral or what
Step-by-step explanation:
Answer:
D) V= 1/6 bh
Step-by-step explanation:
The expression that provides the volume of the pyramid is as follows;
As mentioned in the question
It is given that the pyramid put inside a prism
Also the pyramid contains the similar base area but the height is half of the prism
so the base area of the pyramid is b
And, the Height of the pyramid = h ÷ 2
Now as we know that
The volume of the pyramid is
V = 1 ÷3 × base ares × height
= 1 ÷ 3 × b × h ÷ 2
V = 1 ÷ 6 bh
So, the volume is 1 ÷6 bh cube unit.
Answer:
42 ounces of yellow paint.
Step-by-step explanation:
- To make three cups of dark orange, you just need to remember that to make one cup you need: 7 ounces of red paint and 1 ounce of yellow paint. Then, to make three cups, we just have to multiply these proportions by three,which means 7x3 ounces of red paint =21 ounces of red paint, plus 3x1 ounces of yellow paint =3 ounces of yellow paint. Then we need 21 ounces of red paint and 3 of yellow paint to make three cups of dark orange.
- To make one cup of light orange, we need 13 ounces of yellow paint, and 3 ounces of red paint. Then, to make three cups, we just multiply this proportion by 3, as follows: 13x3 ounces of yellow paint = 39; and 3x3 ounces of red paint =9. Then we need 9 ounces of red paint and 39 of yellow paint to make three cups of light orange.
- Adding up the amount of ounces we need to make dark orange and light orange, we get a total amount of 21+9=30 ounces of red paint and 3+39=42 ounces of yellow paint to make three cups of light orange and three cups of dark orange.
