Answer:
values of k:
any real number works
Step-by-step explanation:
k^2 is always >=0 so
k^2+96 is always >0
Answer: 209
Step-by-step explanation:
Answer:
![k = 1](https://tex.z-dn.net/?f=k%20%3D%201)
![P(x > 3y) = \frac{2}{3}](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cfrac%7B2%7D%7B3%7D)
Step-by-step explanation:
Given
![f \left(x,y \right) = \left{ \begin{array} { l l } { k , } & { 0 \leq x} \leq 2,0 \leq y \leq 1,2 y \leq x } & { \text 0, { elsewhere. } } \end{array} \right.](https://tex.z-dn.net/?f=f%20%5Cleft%28x%2Cy%20%5Cright%29%20%3D%20%5Cleft%7B%20%5Cbegin%7Barray%7D%20%7B%20l%20l%20%7D%20%7B%20k%20%2C%20%7D%20%26%20%7B%200%20%5Cleq%20x%7D%20%5Cleq%202%2C0%20%5Cleq%20y%20%5Cleq%201%2C2%20y%20%20%5Cleq%20x%20%7D%20%20%26%20%7B%20%5Ctext%200%2C%20%7B%20elsewhere.%20%7D%20%7D%20%5Cend%7Barray%7D%20%5Cright.)
Solving (a):
Find k
To solve for k, we use the definition of joint probability function:
![\int\limits^a_b \int\limits^a_b {f(x,y)} \, = 1](https://tex.z-dn.net/?f=%5Cint%5Climits%5Ea_b%20%5Cint%5Climits%5Ea_b%20%7Bf%28x%2Cy%29%7D%20%5C%2C%20%3D%201)
Where
![{ 0 \leq x} \leq 2,0 \leq y \leq 1,2 y \leq x }](https://tex.z-dn.net/?f=%7B%200%20%5Cleq%20x%7D%20%5Cleq%202%2C0%20%5Cleq%20y%20%5Cleq%201%2C2%20y%20%20%5Cleq%20x%20%7D)
Substitute values for the interval of x and y respectively
So, we have:
![\int\limits^2_{0} \int\limits^{x/2}_{0} {k\ dy\ dx} \, = 1](https://tex.z-dn.net/?f=%5Cint%5Climits%5E2_%7B0%7D%20%5Cint%5Climits%5E%7Bx%2F2%7D_%7B0%7D%20%7Bk%5C%20dy%5C%20dx%7D%20%5C%2C%20%3D%201)
Isolate k
![k \int\limits^2_{0} \int\limits^{x/2}_{0} {dy\ dx} \, = 1](https://tex.z-dn.net/?f=k%20%5Cint%5Climits%5E2_%7B0%7D%20%5Cint%5Climits%5E%7Bx%2F2%7D_%7B0%7D%20%7Bdy%5C%20dx%7D%20%5C%2C%20%3D%201)
Integrate y, leave x:
![k \int\limits^2_{0} y {dx} \, [0,x/2]= 1](https://tex.z-dn.net/?f=k%20%5Cint%5Climits%5E2_%7B0%7D%20y%20%7Bdx%7D%20%5C%2C%20%5B0%2Cx%2F2%5D%3D%201)
Substitute 0 and x/2 for y
![k \int\limits^2_{0} (x/2 - 0) {dx} \,= 1](https://tex.z-dn.net/?f=k%20%5Cint%5Climits%5E2_%7B0%7D%20%28x%2F2%20-%200%29%20%7Bdx%7D%20%5C%2C%3D%201)
![k \int\limits^2_{0} \frac{x}{2} {dx} \,= 1](https://tex.z-dn.net/?f=k%20%5Cint%5Climits%5E2_%7B0%7D%20%5Cfrac%7Bx%7D%7B2%7D%20%7Bdx%7D%20%5C%2C%3D%201)
Integrate x
![k * \frac{x^2}{2*2} [0,2]= 1](https://tex.z-dn.net/?f=k%20%2A%20%5Cfrac%7Bx%5E2%7D%7B2%2A2%7D%20%5B0%2C2%5D%3D%201)
![k * \frac{x^2}{4} [0,2]= 1](https://tex.z-dn.net/?f=k%20%2A%20%5Cfrac%7Bx%5E2%7D%7B4%7D%20%5B0%2C2%5D%3D%201)
Substitute 0 and 2 for x
![k *[ \frac{2^2}{4} - \frac{0^2}{4} ]= 1](https://tex.z-dn.net/?f=k%20%2A%5B%20%5Cfrac%7B2%5E2%7D%7B4%7D%20-%20%5Cfrac%7B0%5E2%7D%7B4%7D%20%5D%3D%201)
![k *[ \frac{4}{4} - \frac{0}{4} ]= 1](https://tex.z-dn.net/?f=k%20%2A%5B%20%5Cfrac%7B4%7D%7B4%7D%20-%20%5Cfrac%7B0%7D%7B4%7D%20%5D%3D%201)
![k *[ 1-0 ]= 1](https://tex.z-dn.net/?f=k%20%2A%5B%201-0%20%5D%3D%201)
![k *[ 1]= 1](https://tex.z-dn.net/?f=k%20%2A%5B%201%5D%3D%201)
![k = 1](https://tex.z-dn.net/?f=k%20%3D%201)
Solving (b): ![P(x > 3y)](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29)
We have:
![f(x,y) = k](https://tex.z-dn.net/?f=f%28x%2Cy%29%20%3D%20k)
Where ![k = 1](https://tex.z-dn.net/?f=k%20%3D%201)
![f(x,y) = 1](https://tex.z-dn.net/?f=f%28x%2Cy%29%20%3D%201)
To find
, we use:
![\int\limits^a_b \int\limits^a_b {f(x,y)}](https://tex.z-dn.net/?f=%5Cint%5Climits%5Ea_b%20%5Cint%5Climits%5Ea_b%20%7Bf%28x%2Cy%29%7D)
So, we have:
![P(x > 3y) = \int\limits^2_0 \int\limits^{y/3}_0 {f(x,y)} dxdy](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cint%5Climits%5E2_0%20%5Cint%5Climits%5E%7By%2F3%7D_0%20%7Bf%28x%2Cy%29%7D%20dxdy)
![P(x > 3y) = \int\limits^2_0 \int\limits^{y/3}_0 {1} dxdy](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cint%5Climits%5E2_0%20%5Cint%5Climits%5E%7By%2F3%7D_0%20%7B1%7D%20dxdy)
![P(x > 3y) = \int\limits^2_0 \int\limits^{y/3}_0 dxdy](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cint%5Climits%5E2_0%20%5Cint%5Climits%5E%7By%2F3%7D_0%20%20dxdy)
Integrate x leave y
![P(x > 3y) = \int\limits^2_0 x [0,y/3]dy](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cint%5Climits%5E2_0%20%20x%20%5B0%2Cy%2F3%5Ddy)
Substitute 0 and y/3 for x
![P(x > 3y) = \int\limits^2_0 [y/3 - 0]dy](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cint%5Climits%5E2_0%20%20%5By%2F3%20-%200%5Ddy)
![P(x > 3y) = \int\limits^2_0 y/3\ dy](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cint%5Climits%5E2_0%20%20y%2F3%5C%20dy)
Integrate
![P(x > 3y) = \frac{y^2}{2*3} [0,2]](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cfrac%7By%5E2%7D%7B2%2A3%7D%20%5B0%2C2%5D)
![P(x > 3y) = \frac{y^2}{6} [0,2]\\](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cfrac%7By%5E2%7D%7B6%7D%20%5B0%2C2%5D%5C%5C)
Substitute 0 and 2 for y
![P(x > 3y) = \frac{2^2}{6} -\frac{0^2}{6}](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cfrac%7B2%5E2%7D%7B6%7D%20-%5Cfrac%7B0%5E2%7D%7B6%7D)
![P(x > 3y) = \frac{4}{6} -\frac{0}{6}](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cfrac%7B4%7D%7B6%7D%20-%5Cfrac%7B0%7D%7B6%7D)
![P(x > 3y) = \frac{4}{6}](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cfrac%7B4%7D%7B6%7D)
![P(x > 3y) = \frac{2}{3}](https://tex.z-dn.net/?f=P%28x%20%3E%203y%29%20%3D%20%5Cfrac%7B2%7D%7B3%7D)
Answer:
stress on the cable is 61.312mpa
If steel is used, extension will be 0.91968mm
If polypropylene is used, extension will be 0.1532mm
Step-by-step explanation: detailed explanation and calculation is shown in the image below
Answer:
B
Step-by-step explanation:
First calculate BD using sine ratio in Δ BCD and the exact value
sin60° =
, thus
sin60° =
=
=
=
( cross- multiply )
2BD = 12
( divide both sides by 2 )
BD = 6![\sqrt{3}](https://tex.z-dn.net/?f=%5Csqrt%7B3%7D)
-----------------------------------------------------------
Calculate AD using the tangent ratio in Δ ABD and the exact value
tan30° =
, thus
tan30° =
=
=
=
( cross- multiply )
AD = 6
( divide both sides by
)
AD = 6 → B