Answer:
slope = 
Step-by-step explanation:
Calculate the slope m using the slope formula
m = (y₂ - y₁ ) / (x₂ - x₁ )
with (x₁, y₁ ) = (1, 2) and (x₂, y₂ ) = (5, 4)
m = 
= 
=
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,
The trigonometric equation <span> (sin Θ − cos Θ)^2 − (sin Θ + cos Θ)^3 can be simplified by:
</span>Using x for Θ:
<span>(sinx - cosx)^2 - (sinx + cosx)^2 </span>
<span>= (sin^2 x - 2sinxcosx + cos^2 x) - (sin^2 x + 2sinxcosx + cos^2 x) </span>
<span>= - 2 sinx cosx - 2 sinx cosx </span>
<span>= - 4 sinx cosx </span>
<span>= - 2sin(2x)
</span>
I hope it has come to your help.
Answer:
H0: There is no association between state and sporting preference.
H1: There is an association between state and sporting preference
Step-by-step explanation:
The hypothesis to be tested for is whether the factor 'state' is associated with the factor 'sporting preference'.
The study is therefore about 'association' and whether the distributions of sporting preferences are identical across states. In scenario in this case is the test for association which is the most appropriate test.
Two factors are deemed to not be associated unless there is supporting evidence to suggest otherwise. Since the null hypothesis is the default belief, the correct pair of hypotheses are:
H0: There is no association between state and sporting preference.
H1: There is an association between state and sporting preference
A'B'C'D after the translation. Corresponding parts of congruent figures are congruent. Angle BCD corresponds to angle B'C'D', so angle BCD is congruent to angle B'C'D'
See attached. The graph confirms this.
