Answer:
![A=\pi\displaystyle\biggr[\frac{16}{3}-2\ln(|3|)\biggr]\approx9.8524](https://tex.z-dn.net/?f=A%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B%5Cfrac%7B16%7D%7B3%7D-2%5Cln%28%7C3%7C%29%5Cbiggr%5D%5Capprox9.8524)
Step-by-step explanation:
Use the Washer Method ![A=\pi\displaystyle \int\limits_{a}^{b}{\bigr[R(x)^2-r(x)^2\bigr] \, dx](https://tex.z-dn.net/?f=A%3D%5Cpi%5Cdisplaystyle%20%5Cint%5Climits_%7Ba%7D%5E%7Bb%7D%7B%5Cbigr%5BR%28x%29%5E2-r%28x%29%5E2%5Cbigr%5D%20%5C%2C%20dx)
where 
is the outer radius and 
is the inner radius.
If we sketch out the graph, we see that 
intersects points 
and 
, which will be our bounds of integration.
Here, our outer radius will be 
and our inner radius will be 
.
Thus, we can compute the integral and find the volume:
![A=\pi\displaystyle\int\limits^{3}_{1} {(-2)^2-\biggr(-1-\frac{1}{x}\biggr)^2 } \, dx\\ \\A=\pi\displaystyle\int\limits^{3}_{1} {4-\biggr(1+\frac{2}{x}+\frac{1}{x^2} \biggr) } \, dx\\\\A=\pi\displaystyle\int\limits^{3}_{1} {4-1-\frac{2}{x}-\frac{1}{x^2}} \, dx\\\\A=\pi\displaystyle\int\limits^{3}_{1} {3-\frac{2}{x}-\frac{1}{x^2}} \, dx\\\\A=\pi\displaystyle\biggr[3x-2\ln(|x|)+\frac{1}{x}\biggr]\Biggr|_{1}^{3}\\](https://tex.z-dn.net/?f=A%3D%5Cpi%5Cdisplaystyle%5Cint%5Climits%5E%7B3%7D_%7B1%7D%20%7B%28-2%29%5E2-%5Cbiggr%28-1-%5Cfrac%7B1%7D%7Bx%7D%5Cbiggr%29%5E2%20%7D%20%5C%2C%20dx%5C%5C%20%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cint%5Climits%5E%7B3%7D_%7B1%7D%20%7B4-%5Cbiggr%281%2B%5Cfrac%7B2%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%20%20%5Cbiggr%29%20%7D%20%5C%2C%20dx%5C%5C%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cint%5Climits%5E%7B3%7D_%7B1%7D%20%7B4-1-%5Cfrac%7B2%7D%7Bx%7D-%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%20%5C%2C%20dx%5C%5C%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cint%5Climits%5E%7B3%7D_%7B1%7D%20%7B3-%5Cfrac%7B2%7D%7Bx%7D-%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%20%5C%2C%20dx%5C%5C%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B3x-2%5Cln%28%7Cx%7C%29%2B%5Cfrac%7B1%7D%7Bx%7D%5Cbiggr%5D%5CBiggr%7C_%7B1%7D%5E%7B3%7D%5C%5C)
![A=\pi\displaystyle\biggr[\biggr(3(3)-2\ln(|3|)+\frac{1}{3}\biggr)-\biggr(3(1)-2\ln(|1|)+\frac{1}{1}\biggr)\biggr]\\\\A=\pi\displaystyle\biggr[\biggr(9-2\ln(|3|)+\frac{1}{3}\biggr)-\biggr(3+1\biggr)\biggr]\\\\A=\pi\displaystyle\biggr[\biggr(\frac{28}{3}-2\ln(|3|)\biggr)-\biggr(4\biggr)\biggr]\\A=\pi\displaystyle\biggr[\frac{16}{3}-2\ln(|3|)\biggr]\\A\approx9.8524](https://tex.z-dn.net/?f=A%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B%5Cbiggr%283%283%29-2%5Cln%28%7C3%7C%29%2B%5Cfrac%7B1%7D%7B3%7D%5Cbiggr%29-%5Cbiggr%283%281%29-2%5Cln%28%7C1%7C%29%2B%5Cfrac%7B1%7D%7B1%7D%5Cbiggr%29%5Cbiggr%5D%5C%5C%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B%5Cbiggr%289-2%5Cln%28%7C3%7C%29%2B%5Cfrac%7B1%7D%7B3%7D%5Cbiggr%29-%5Cbiggr%283%2B1%5Cbiggr%29%5Cbiggr%5D%5C%5C%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B%5Cbiggr%28%5Cfrac%7B28%7D%7B3%7D-2%5Cln%28%7C3%7C%29%5Cbiggr%29-%5Cbiggr%284%5Cbiggr%29%5Cbiggr%5D%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B%5Cfrac%7B16%7D%7B3%7D-2%5Cln%28%7C3%7C%29%5Cbiggr%5D%5C%5CA%5Capprox9.8524)
In conclusion, the volume of the solid of revolution will be about 9.8524 cubic units. See the attached graph for a helpful visual!
Answer:
3xyz
Step-by-step explanation:
The required sum = 7xyz+(−5xyz)+9xyz+(−8xyz)
=7xyz–5xyz+9xyz–8xyz
=(7–5+9–8)xyz
=(16–13)xyz
=3xyz
Answer:
(-∞,0)∪(0,∞)
Step-by-step explanation:
The domain of a function is wherever the y values exist. The only time the y values dont exist in this function is when the denominator is zero. So the denominator does not exist at x=0, because that is when 4x = 0.
R(p) = -15p² + 200p + 10,000
To find the maximum of the function, we take the first derivative and equate to 0:
R'(p) = -30p + 200
0 = -30p + 200
p = 6.67
A price of $6.67 will maximize revenue.
Answer:
Total number of tables of first type = 23.
Total number of tables of second type = 7
Step-by-step explanation:
It is given that there are 30 tables in total and there are two types of tables.
Let's call the two seat tables, the first type as x and the second type as y.
∴ x + y = 30 ......(1)
Also a total number of 81 people are seated. Therefore, 2x number of people would be seated on the the first type and 5y on the second type. Hence the equation becomes:
2x + 5y = 81 .....(2)
To solve (1) & (2) Multiply (1) by 2 and subtract, we get:
y = 7
Substituting y = 7 in (1), we get x = 23.
∴ The number of tables of first kind = 23
The number of tables of second kind = 7
