The dimensions of a cylinder which has this maximum volume are equal to 1.83 units and 3 units.
<u>Given the following data:</u>
- Height of cylinder, h = 5.5 units.
- Radius of cylinder, r = 4.5 units.
<h3>How to calculate the volume of this cylinder?</h3>
Mathematically, the volume of a cylinder can be calculated by using this formula:
V = πr²h
Next, we would convert the above multi-variable function into a single-variable function by applying the properties of 2 similar triangles:
H/H - h = R/r
H - h = r(H/R)
h = H/R(R - r)
V = πHr²/R(R - r)
In order to maximize the volume of this cylinder, we would determine the critical points of the function by differentiating wrt r:
dV/dr = πH/R(2rR - 2r² - r²)
(2rR - 3r²) = 0
r = 2R/3
r = (2 × 4.5)/3
Maximum radius, r = 3 units.
For the max. height, we have:
h = H/R(R - r)
h = H/R(R - 2R/3)
h = H/3
h = 5.5/3
Maximum height, h = 1.83 units.
Read more on cylinder here: brainly.com/question/315709
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Answer:
36 is dividend
in 36/6, 6 is the divisor
the other 6 is quotient
Step-by-step explanation:
<u>Answer:</u>
![y = - \frac{1}{24} (x + 5) + 1](https://tex.z-dn.net/?f=y%20%3D%20%20-%20%20%5Cfrac%7B1%7D%7B24%7D%20%28x%20%2B%205%29%20%2B%201)
<u>Explanation</u>
The directrix y=7, is above the y-value of the focus. The parabola must will open downwards.
Such parabola has equation of the form,
![{(x - h)}^{2} = - 4p(y - k)](https://tex.z-dn.net/?f=%20%7B%28x%20-%20h%29%7D%5E%7B2%7D%20%20%3D%20-%20%204p%28y%20-%20k%29)
where (h,k) is the vertex.
The vertex is the midway from the focus to the directrix
The x-value of the vertex is x=-5 because it is on a vertical line that goes through (-5,-5).
The y-value of the vertex is
![y = \frac{ 7 + - 5}{2}](https://tex.z-dn.net/?f=y%20%3D%20%20%5Cfrac%7B%207%20%2B%20%20-%205%7D%7B2%7D%20)
![y = \frac{ 2}{2} = 1](https://tex.z-dn.net/?f=y%20%3D%20%20%5Cfrac%7B%202%7D%7B2%7D%20%20%3D%20%201)
The equation of the parabola now becomes
![{(x + 5)}^{2} = - 4p(y - 1)](https://tex.z-dn.net/?f=%7B%28x%20%20%2B%205%29%7D%5E%7B2%7D%20%20%3D%20-%20%204p%28y%20%20%20-%201%29)
p is the distance from the focus to the vertex which is p=|7-1|=6
Substitute the value of p to get:
![{(x + 5)}^{2} = - 4 \times 6(y - 1)](https://tex.z-dn.net/?f=%7B%28x%20%20%2B%205%29%7D%5E%7B2%7D%20%20%3D%20-%20%204%20%5Ctimes%206%28y%20%20%20-%201%29)
![{(x + 5)}^{2} = - 24(y - 1)](https://tex.z-dn.net/?f=%7B%28x%20%20%2B%205%29%7D%5E%7B2%7D%20%20%3D%20-%2024%28y%20%20%20-%201%29)
We solve for y to get:
![y = - \frac{1}{24} (x + 5) + 1](https://tex.z-dn.net/?f=y%20%3D%20%20-%20%20%5Cfrac%7B1%7D%7B24%7D%20%28x%20%2B%205%29%20%2B%201)