9y+7=2y+98
9y-2y=98-7
7y=91 /:7
y=13
(9y+7)°=(9·13+7)°=(117+7)°=124°
Answer:
697cm³
Step-by-step explanation:
Volume = Base area* Height
= 17 x 10* 4.1
= 697cm³
Answer:
The mean of the cable length is ![\mu=1205](https://tex.z-dn.net/?f=%5Cmu%3D1205)
The standard deviation of the cable length is ![\sigma=2.886](https://tex.z-dn.net/?f=%5Csigma%3D2.886)
Half of the cables lie in the specifications.
Step-by-step explanation:
<em>Point a:</em>
Suppose <em>X</em> is a continuous random variable with probability density function <em>f(x).</em>
The mean of a continuous random variable, denoted as <em>μ</em> or <em>E(X)</em> is<em> </em>
<em>
</em>
The standard deviation of <em>X</em> is
![\sigma=\sqrt{\sigma^2}](https://tex.z-dn.net/?f=%5Csigma%3D%5Csqrt%7B%5Csigma%5E2%7D)
where
is the variance of X
![\sigma^2=\int\limits^\infty_{-\infty} {x^2f(x)} \, dx-\mu^2](https://tex.z-dn.net/?f=%5Csigma%5E2%3D%5Cint%5Climits%5E%5Cinfty_%7B-%5Cinfty%7D%20%7Bx%5E2f%28x%29%7D%20%5C%2C%20dx-%5Cmu%5E2)
We know that the probability density function of the length of computer cables is
![f(x)=0.1, \:1200](https://tex.z-dn.net/?f=f%28x%29%3D0.1%2C%20%5C%3A1200%3Cx%3C1210)
Applying the above definition of the mean we get
![E(X)=\int\limits^{1210}_{1200} {0.1x} \, dx =1205\\\\\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx\\\\0.1\cdot \int _{1200}^{1210}xdx\\\\\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1\\\\0.1\left[\frac{x^{1+1}}{1+1}\right]^{1210}_{1200}\\\\Simplify\\\\0.1\left[0.5x^2\right]^{1210}_{1200}\\\\\mathrm{Compute\:the\:boundaries}:\quad \left[0.5x^2\right]^{1210}_{1200}=12050\\\\0.1\cdot \:12050=1205](https://tex.z-dn.net/?f=E%28X%29%3D%5Cint%5Climits%5E%7B1210%7D_%7B1200%7D%20%7B0.1x%7D%20%5C%2C%20dx%20%3D1205%5C%5C%5C%5C%5Cmathrm%7BTake%5C%3Athe%5C%3Aconstant%5C%3Aout%7D%3A%5Cquad%20%5Cint%20a%5Ccdot%20f%5Cleft%28x%5Cright%29dx%3Da%5Ccdot%20%5Cint%20f%5Cleft%28x%5Cright%29dx%5C%5C%5C%5C0.1%5Ccdot%20%5Cint%20_%7B1200%7D%5E%7B1210%7Dxdx%5C%5C%5C%5C%5Cmathrm%7BApply%5C%3Athe%5C%3APower%5C%3ARule%7D%3A%5Cquad%20%5Cint%20x%5Eadx%3D%5Cfrac%7Bx%5E%7Ba%2B1%7D%7D%7Ba%2B1%7D%2C%5C%3A%5Cquad%20%5C%3Aa%5Cne%20-1%5C%5C%5C%5C0.1%5Cleft%5B%5Cfrac%7Bx%5E%7B1%2B1%7D%7D%7B1%2B1%7D%5Cright%5D%5E%7B1210%7D_%7B1200%7D%5C%5C%5C%5CSimplify%5C%5C%5C%5C0.1%5Cleft%5B0.5x%5E2%5Cright%5D%5E%7B1210%7D_%7B1200%7D%5C%5C%5C%5C%5Cmathrm%7BCompute%5C%3Athe%5C%3Aboundaries%7D%3A%5Cquad%20%5Cleft%5B0.5x%5E2%5Cright%5D%5E%7B1210%7D_%7B1200%7D%3D12050%5C%5C%5C%5C0.1%5Ccdot%20%5C%3A12050%3D1205)
Applying the above definition of the standard deviation we get
First we need to calculate the variance of X
![\sigma^2=\int\limits^{1210}_{1200} {x^2\cdot 0.1} \, dx-\mu^2](https://tex.z-dn.net/?f=%5Csigma%5E2%3D%5Cint%5Climits%5E%7B1210%7D_%7B1200%7D%20%7Bx%5E2%5Ccdot%200.1%7D%20%5C%2C%20dx-%5Cmu%5E2)
![\int _{1200}^{1210}0.1x^2dx\\\\\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx\\\\0.1\cdot \int _{1200}^{1210}x^2dx\\\\\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1\\\\0.1\left[\frac{x^{2+1}}{2+1}\right]^{1210}_{1200}\\\\0.1\left[\frac{1}{3}\cdot x^3\right]^{1210}_{1200}\\\\\mathrm{Compute\:the\:boundaries}:\quad \left[\frac{1}{3}\cdot x^3\right]^{1210}_{1200}=14520333.33\\\\0.1\cdot \:14520333.33=1452033.33](https://tex.z-dn.net/?f=%5Cint%20_%7B1200%7D%5E%7B1210%7D0.1x%5E2dx%5C%5C%5C%5C%5Cmathrm%7BTake%5C%3Athe%5C%3Aconstant%5C%3Aout%7D%3A%5Cquad%20%5Cint%20a%5Ccdot%20f%5Cleft%28x%5Cright%29dx%3Da%5Ccdot%20%5Cint%20f%5Cleft%28x%5Cright%29dx%5C%5C%5C%5C0.1%5Ccdot%20%5Cint%20_%7B1200%7D%5E%7B1210%7Dx%5E2dx%5C%5C%5C%5C%5Cmathrm%7BApply%5C%3Athe%5C%3APower%5C%3ARule%7D%3A%5Cquad%20%5Cint%20x%5Eadx%3D%5Cfrac%7Bx%5E%7Ba%2B1%7D%7D%7Ba%2B1%7D%2C%5C%3A%5Cquad%20%5C%3Aa%5Cne%20-1%5C%5C%5C%5C0.1%5Cleft%5B%5Cfrac%7Bx%5E%7B2%2B1%7D%7D%7B2%2B1%7D%5Cright%5D%5E%7B1210%7D_%7B1200%7D%5C%5C%5C%5C0.1%5Cleft%5B%5Cfrac%7B1%7D%7B3%7D%5Ccdot%20x%5E3%5Cright%5D%5E%7B1210%7D_%7B1200%7D%5C%5C%5C%5C%5Cmathrm%7BCompute%5C%3Athe%5C%3Aboundaries%7D%3A%5Cquad%20%5Cleft%5B%5Cfrac%7B1%7D%7B3%7D%5Ccdot%20x%5E3%5Cright%5D%5E%7B1210%7D_%7B1200%7D%3D14520333.33%5C%5C%5C%5C0.1%5Ccdot%20%5C%3A14520333.33%3D1452033.33)
![\sigma^2=1452033.33-1205^2=8.33](https://tex.z-dn.net/?f=%5Csigma%5E2%3D1452033.33-1205%5E2%3D8.33)
![\sigma=\sqrt{\sigma^2}=\sqrt{8.33}=2.886](https://tex.z-dn.net/?f=%5Csigma%3D%5Csqrt%7B%5Csigma%5E2%7D%3D%5Csqrt%7B8.33%7D%3D2.886)
<em>Point b:</em>
To find what proportion of cables is within specifications you need to:
![P(1195](https://tex.z-dn.net/?f=P%281195%3CX%3C1205%29%3D%5Cint%5Climits%5E%7B1200%7D_%7B1195%7D%20%7B0%7D%20%5C%2C%20dx%20%2B%20%5Cint%5Climits%5E%7B1205%7D_%7B1200%7D%20%7B0.1%7D%20%5C%2C%20dx%5C%5CP%281195%3CX%3C1205%29%3D0%2B0.5%3D0.5)
Answer:
Multiply each equation by the value that makes the coefficients of
x
opposite.
(
−
3
)
⋅
(
2
x
+
5
y
)
=
(
−
3
)
(
20
)
(
2
)
⋅
(
3
x
−
10
y
)
=
(
2
)
(
37
)
Step-by-step explanation:
Answer:
Step-by-step explanation:
If we divide 24 by 6, we get 4, which is the other factor of 24 when given the one factor of 6. Same goes here. In order to find out what the other factor of
is when given one factor of y - 4, we simply divide the second degree polynomial by y - 4 to get the quotient. The quotient, then, is the other factor. Synthetic division is the easiest way to do this.
If y - 4 is the factor, then by the Zero Product Property, y - 4 = 0 and y = 4. Setting up synthetic division:
4| 1 -10 24
____________
The rule is to start by bringing down the first term, multiplying it by the number outside, then putting that product up under the next term in line:
4| 1 -10 24
<u> 4 </u>
1
Then add the column, multiply the sum by the number outside, and put that product up under the next term in line:
4| 1 -10 24
<u> 4 -24</u>
1 -6
And add the last column and that is the remainder. We get a 0 remainder. That means that y - 4 goes evenly into the polynomial and the other factor we are looking for is found in the numbers under the addition line. These numbers are the leading coefficients of the depressed polynomial, the polynomial that serves as the other factor: 1y - 6.
Therefore, the 2 factors that multiply together to give us
are (y - 4)(y - 6) and we can check ourselves by multiplying this out by FOILing to see if the result is the polynomial we started with. It is, so we're all done!