Answer:
n = 6
, −
6
Step-by-step explanation:
Take the root of both sides and solve.
Answer:
To find the length of a horizontal line segment, find the difference between the x-coordinates. Subtract the smaller from the larger.
72/3=24
24/12=2(2 meters a person a day)
2*15=30
15 people can build about 30 meters of fence a day.
Answer:
The exterior angle is 77
Step-by-step explanation:
The measure of the exterior angle of a triangle is equal to the sum of the two opposite interior angles.
2x+11 = x +44
First we need to find x
Subtract x from each side
2x+11-x = x+44
x +11 = 44
Subtract 11 from each side
x+11-11 = 44-11
x =33
Now we can find the exterior angle
2x+11 = 2(33)+11 = 66+11 = 77
The exterior angle is 77
It looks like the boundaries of
are the lines
and
, as well as the hyperbolas
and
. Naturally, the domain of integration is the set
![R = \left\{(x,y) ~:~ \dfrac{2x}3 \le y \le 3x \text{ and } \dfrac23 \le xy \le 3 \right\}](https://tex.z-dn.net/?f=R%20%3D%20%5Cleft%5C%7B%28x%2Cy%29%20~%3A~%20%5Cdfrac%7B2x%7D3%20%5Cle%20y%20%5Cle%203x%20%5Ctext%7B%20and%20%7D%20%5Cdfrac23%20%5Cle%20xy%20%5Cle%203%20%5Cright%5C%7D)
By substituting
and
, so
, we have
![\dfrac23 \le xy \le 3 \implies \dfrac23 \le u \le 3](https://tex.z-dn.net/?f=%5Cdfrac23%20%5Cle%20xy%20%5Cle%203%20%5Cimplies%20%5Cdfrac23%20%5Cle%20u%20%5Cle%203)
and
![\dfrac{2x}3 \le y \le 3x \implies \dfrac{2u}{3v} \le v \le \dfrac{3u}v \implies \dfrac{2u}3 \le v^2 \le 3u \implies \sqrt{\dfrac{2u}3} \le v \le \sqrt{3u}](https://tex.z-dn.net/?f=%5Cdfrac%7B2x%7D3%20%5Cle%20y%20%5Cle%203x%20%5Cimplies%20%5Cdfrac%7B2u%7D%7B3v%7D%20%5Cle%20v%20%5Cle%20%5Cdfrac%7B3u%7Dv%20%5Cimplies%20%5Cdfrac%7B2u%7D3%20%5Cle%20v%5E2%20%5Cle%203u%20%5Cimplies%20%5Csqrt%7B%5Cdfrac%7B2u%7D3%7D%20%5Cle%20v%20%5Cle%20%5Csqrt%7B3u%7D)
so that
![R = \left\{(u,v) ~:~ \dfrac23 \le u \le 3 \text{ and } \sqrt{\dfrac{2u}3 \le v \le \sqrt{3u}\right\}](https://tex.z-dn.net/?f=R%20%3D%20%5Cleft%5C%7B%28u%2Cv%29%20~%3A~%20%5Cdfrac23%20%5Cle%20u%20%5Cle%203%20%5Ctext%7B%20and%20%7D%20%5Csqrt%7B%5Cdfrac%7B2u%7D3%20%5Cle%20v%20%5Cle%20%5Csqrt%7B3u%7D%5Cright%5C%7D)
Compute the Jacobian for this transformation and its determinant.
![J = \begin{bmatrix}x_u & x_v \\ y_u & y_v\end{bmatrix} = \begin{bmatrix}\dfrac1v & -\dfrac u{v^2} \\\\ 0 & 1 \end{bmatrix} \implies \det(J) = \dfrac1v](https://tex.z-dn.net/?f=J%20%3D%20%5Cbegin%7Bbmatrix%7Dx_u%20%26%20x_v%20%5C%5C%20y_u%20%26%20y_v%5Cend%7Bbmatrix%7D%20%3D%20%5Cbegin%7Bbmatrix%7D%5Cdfrac1v%20%26%20-%5Cdfrac%20u%7Bv%5E2%7D%20%5C%5C%5C%5C%200%20%26%201%20%5Cend%7Bbmatrix%7D%20%5Cimplies%20%5Cdet%28J%29%20%3D%20%5Cdfrac1v)
Then the area element under this change of variables is
![dA = dx\,dy = \dfrac{du\,dv}v](https://tex.z-dn.net/?f=dA%20%3D%20dx%5C%2Cdy%20%3D%20%5Cdfrac%7Bdu%5C%2Cdv%7Dv)
and the integral transforms to
![\displaystyle \iint_R 9xy \, dA = \int_{2/3}^3 \int_{\sqrt{2u/3}}^{\sqrt{3u}} \frac{dv\,du}v](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ciint_R%209xy%20%5C%2C%20dA%20%3D%20%5Cint_%7B2%2F3%7D%5E3%20%5Cint_%7B%5Csqrt%7B2u%2F3%7D%7D%5E%7B%5Csqrt%7B3u%7D%7D%20%5Cfrac%7Bdv%5C%2Cdu%7Dv)
Now compute it.
![\displaystyle \iint_R 9xy \, dA = \int_{2/3}^3 \ln|v|\bigg|_{v=\sqrt{2u/3}}^{v=\sqrt{3u}} \,du \\\\ ~~~~~~~~ = \int_{2/3}^3 \ln\left(\sqrt{3u}\right) - \ln\left(\sqrt{\frac{2u}3}\right) \, du \\\\ ~~~~~~~~ = \frac12 \int_{2/3}^3 \ln(3u) - \ln\left(\frac{2u}3\right) \, du \\\\ ~~~~~~~~ = \frac12 \int_{2/3}^3 \ln\left(\frac{3u}{\frac{2u}3}\right) \, du \\\\ ~~~~~~~~ = \frac12 \ln\left(\frac92\right) \int_{2/3}^3 du \\\\ ~~~~~~~~ = \frac12 \ln\left(\frac92\right) \left(3-\frac23\right) = \boxed{\frac76 \ln\left(\frac92\right)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ciint_R%209xy%20%5C%2C%20dA%20%3D%20%5Cint_%7B2%2F3%7D%5E3%20%5Cln%7Cv%7C%5Cbigg%7C_%7Bv%3D%5Csqrt%7B2u%2F3%7D%7D%5E%7Bv%3D%5Csqrt%7B3u%7D%7D%20%5C%2Cdu%20%5C%5C%5C%5C%20~~~~~~~~%20%3D%20%5Cint_%7B2%2F3%7D%5E3%20%5Cln%5Cleft%28%5Csqrt%7B3u%7D%5Cright%29%20-%20%5Cln%5Cleft%28%5Csqrt%7B%5Cfrac%7B2u%7D3%7D%5Cright%29%20%5C%2C%20du%20%5C%5C%5C%5C%20~~~~~~~~%20%3D%20%5Cfrac12%20%5Cint_%7B2%2F3%7D%5E3%20%5Cln%283u%29%20-%20%5Cln%5Cleft%28%5Cfrac%7B2u%7D3%5Cright%29%20%5C%2C%20du%20%5C%5C%5C%5C%20~~~~~~~~%20%3D%20%5Cfrac12%20%5Cint_%7B2%2F3%7D%5E3%20%5Cln%5Cleft%28%5Cfrac%7B3u%7D%7B%5Cfrac%7B2u%7D3%7D%5Cright%29%20%5C%2C%20du%20%5C%5C%5C%5C%20~~~~~~~~%20%3D%20%5Cfrac12%20%5Cln%5Cleft%28%5Cfrac92%5Cright%29%20%5Cint_%7B2%2F3%7D%5E3%20du%20%5C%5C%5C%5C%20~~~~~~~~%20%3D%20%5Cfrac12%20%5Cln%5Cleft%28%5Cfrac92%5Cright%29%20%5Cleft%283-%5Cfrac23%5Cright%29%20%3D%20%5Cboxed%7B%5Cfrac76%20%5Cln%5Cleft%28%5Cfrac92%5Cright%29%7D)