Answer:
a) ![P(X = 1) = 0.38742](https://tex.z-dn.net/?f=P%28X%20%3D%201%29%20%3D%200.38742)
b) ![P(X = 3) = 0.05740](https://tex.z-dn.net/?f=P%28X%20%3D%203%29%20%3D%200.05740)
c) ![P(X = 9) = 0.00000](https://tex.z-dn.net/?f=P%28X%20%3D%209%29%20%3D%200.00000)
d) ![P(X \geq 5) = 0.00163](https://tex.z-dn.net/?f=P%28X%20%5Cgeq%205%29%20%3D%200.00163)
Step-by-step explanation:
For each container, there are only two possible outcomes. Either it is undefilled, or it is not. This means that we can solve this problem using the binomial probability distribution.
Binomial probability distribution:
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
In which
is the number of different combinatios of x objects from a set of n elements, given by the following formula.
![C_{n,x} = \frac{n!}{x!(n-x)!}](https://tex.z-dn.net/?f=C_%7Bn%2Cx%7D%20%3D%20%5Cfrac%7Bn%21%7D%7Bx%21%28n-x%29%21%7D)
And p is the probability of X happening.
In this problem
There are 10 containers, so
.
A food-packaging apparatus underfills 10% of the containers, so
.
a) This is P(X = 1)
![P(X = 1) = C_{10,1}.(0.1)^{1}.(0.9)^{9} = 0.38742](https://tex.z-dn.net/?f=P%28X%20%3D%201%29%20%3D%20C_%7B10%2C1%7D.%280.1%29%5E%7B1%7D.%280.9%29%5E%7B9%7D%20%3D%200.38742)
b) This is P(X = 3)
![P(X = 3) = C_{10,3}.(0.1)^{3}.(0.9)^{7} = 0.05740](https://tex.z-dn.net/?f=P%28X%20%3D%203%29%20%3D%20C_%7B10%2C3%7D.%280.1%29%5E%7B3%7D.%280.9%29%5E%7B7%7D%20%3D%200.05740)
c) This is P(X = 9)
![P(X = 9) = C_{10,9}.(0.1)^{9}.(0.9)^{1} = 0.00000](https://tex.z-dn.net/?f=P%28X%20%3D%209%29%20%3D%20C_%7B10%2C9%7D.%280.1%29%5E%7B9%7D.%280.9%29%5E%7B1%7D%20%3D%200.00000)
d) This is
.
Either the number is lesser than five, or it is five or larger. The sum of the probabilities of each event is decimal 1. So:
![P(X < 5) + P(X \geq 5) = 1](https://tex.z-dn.net/?f=P%28X%20%3C%205%29%20%2B%20P%28X%20%5Cgeq%205%29%20%3D%201)
![P(X \geq 5) = 1 - P(X < 5)](https://tex.z-dn.net/?f=P%28X%20%5Cgeq%205%29%20%3D%201%20-%20P%28X%20%3C%205%29)
In which
![P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)](https://tex.z-dn.net/?f=P%28X%20%3C%205%29%20%3D%20P%28X%20%3D%200%29%20%2B%20P%28X%20%3D%201%29%20%2B%20P%28X%20%3D%202%29%20%2B%20P%28X%20%3D%203%29%20%2B%20P%28X%20%3D%204%29)
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 0) = C_{10,0}.(0.1)^{0}.(0.9)^{10} = 0.34868](https://tex.z-dn.net/?f=P%28X%20%3D%200%29%20%3D%20C_%7B10%2C0%7D.%280.1%29%5E%7B0%7D.%280.9%29%5E%7B10%7D%20%3D%200.34868)
![P(X = 1) = C_{10,1}.(0.1)^{1}.(0.9)^{9} = 0.38742](https://tex.z-dn.net/?f=P%28X%20%3D%201%29%20%3D%20C_%7B10%2C1%7D.%280.1%29%5E%7B1%7D.%280.9%29%5E%7B9%7D%20%3D%200.38742)
![P(X = 2) = C_{10,2}.(0.1)^{2}.(0.9)^{8} = 0.1937](https://tex.z-dn.net/?f=P%28X%20%3D%202%29%20%3D%20C_%7B10%2C2%7D.%280.1%29%5E%7B2%7D.%280.9%29%5E%7B8%7D%20%3D%200.1937)
![P(X = 3) = C_{10,3}.(0.1)^{3}.(0.9)^{7} = 0.05740](https://tex.z-dn.net/?f=P%28X%20%3D%203%29%20%3D%20C_%7B10%2C3%7D.%280.1%29%5E%7B3%7D.%280.9%29%5E%7B7%7D%20%3D%200.05740)
![P(X = 4) = C_{10,4}.(0.1)^{1}.(0.9)^{9} = 0.38742](https://tex.z-dn.net/?f=P%28X%20%3D%204%29%20%3D%20C_%7B10%2C4%7D.%280.1%29%5E%7B1%7D.%280.9%29%5E%7B9%7D%20%3D%200.38742)
So
![P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 0.34868 + 0.38742 + 0.19371 + 0.05740 + 0.01116 = 0.99837](https://tex.z-dn.net/?f=P%28X%20%3C%205%29%20%3D%20P%28X%20%3D%200%29%20%2B%20P%28X%20%3D%201%29%20%2B%20P%28X%20%3D%202%29%20%2B%20P%28X%20%3D%203%29%20%2B%20P%28X%20%3D%204%29%20%3D%200.34868%20%2B%200.38742%20%2B%200.19371%20%2B%200.05740%20%2B%200.01116%20%3D%200.99837)
Finally
![P(X \geq 5) = 1 - P(X < 5) = 1 - 0.99837 = 0.00163](https://tex.z-dn.net/?f=P%28X%20%5Cgeq%205%29%20%3D%201%20-%20P%28X%20%3C%205%29%20%3D%201%20-%200.99837%20%3D%200.00163)
1) The function is:
![f(x) = {x}^{2} + 6x - 16](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%20%7Bx%7D%5E%7B2%7D%20%20%20%2B%206x%20-%20%2016)
At x=2,
![f(2) = {2}^{2} + 6(2) - 16 \\ f(2) = 4 + 12 - 16 = 0](https://tex.z-dn.net/?f=f%282%29%20%3D%20%20%7B2%7D%5E%7B2%7D%20%20%20%2B%20%206%282%29%20%20-%2016%20%5C%5C%20f%282%29%20%3D%204%20%20%2B%20%2012%20%20-%2016%20%3D%200)
Find the first derivative,
![f'(x) = 2x + 6](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%202x%20%20%2B%20%206)
Find the slope at x=2
![f'(2) = 2 \times 2 + 6 = 10](https://tex.z-dn.net/?f=f%27%282%29%20%3D%202%20%5Ctimes%202%20%20%2B%206%20%3D%20%2010)
The equation of the tangent line is given by:
![y - f(2) = f'(2)(x - 2)](https://tex.z-dn.net/?f=y%20-%20f%282%29%20%3D%20f%27%282%29%28x%20-%202%29)
![y - 0= 10(x - 2) \\ y = 10x - 20](https://tex.z-dn.net/?f=y%20-%200%3D%20%2010%28x%20-%202%29%20%5C%5C%20y%20%3D%2010x%20%20%20-%2020)
Therefore
![\boxed{f(x) = {x}^{2} - 6x + 16 \to \: y = 10x - 21}](https://tex.z-dn.net/?f=%20%5Cboxed%7Bf%28x%29%20%3D%20%20%7Bx%7D%5E%7B2%7D%20%20%20-%20%206x%20%2B%2016%20%5Cto%20%5C%3A%20y%20%3D%20%2010x%20%20-%2021%7D)
2) The given function is
![g(x) = {x}^{2} - 49x - 456](https://tex.z-dn.net/?f=g%28x%29%20%3D%20%20%7Bx%7D%5E%7B2%7D%20%20-%2049x%20-%20456)
![g(2) = {2}^{2} - 49 \times 2 - 456 = - 550](https://tex.z-dn.net/?f=g%282%29%20%3D%20%20%7B2%7D%5E%7B2%7D%20%20-%2049%20%5Ctimes%202%20-%20456%20%3D%20%20-%20550)
![g'(x) = 2x - 49 \\ g'(2) = 2 \times 2 - 49 = - 45](https://tex.z-dn.net/?f=g%27%28x%29%20%3D%202x%20-%2049%20%5C%5C%20g%27%282%29%20%3D%202%20%5Ctimes%202%20-%2049%20%3D%20%20-%2045)
The equation of the tangent is
![y - g(2) = g'(2)(x - 2) \\ y + 550 = - 45(x - 2) \\ y + 550 = - 45x + 90 \\ y = - 45x + 90 - 550 \\ y = - 45x - 460](https://tex.z-dn.net/?f=y%20-%20g%282%29%20%3D%20g%27%282%29%28x%20-%202%29%20%5C%5C%20y%20%2B%20550%20%3D%20%20-%2045%28x%20-%202%29%20%5C%5C%20y%20%2B%20550%20%3D%20%20-%2045x%20%2B%2090%20%5C%5C%20y%20%3D%20%20-%2045x%20%2B%2090%20-%20550%20%5C%5C%20y%20%3D%20%20-%2045x%20-%20460)
![\boxed{g(x) = {x}^{2} - 45x - 456 \to \: y = - 45x - 460}](https://tex.z-dn.net/?f=%20%5Cboxed%7Bg%28x%29%20%3D%20%20%7Bx%7D%5E%7B2%7D%20%20%20-%20%2045x%20%20-%20456%20%5Cto%20%5C%3A%20y%20%3D%20%20%20-%2045x%20%20-%20460%7D)
3) The function is
![h(x) = - {x}^{2} - 7x + 44](https://tex.z-dn.net/?f=h%28x%29%20%3D%20%20-%20%20%7Bx%7D%5E%7B2%7D%20%20-%207x%20%20%2B%2044)
Now
![h(2) = - {2}^{2} - 7 \times 2 + 44 = 26](https://tex.z-dn.net/?f=h%282%29%20%3D%20%20-%20%20%7B2%7D%5E%7B2%7D%20%20-%207%20%5Ctimes%202%20%20%2B%2044%20%3D%2026)
The first derivative is
![h'(x) = - 2x - 7](https://tex.z-dn.net/?f=h%27%28x%29%20%3D%20%20-%202x%20-%207)
![h'(2) = - 2 \times 2 - 7 = - 14](https://tex.z-dn.net/?f=h%27%282%29%20%3D%20%20-%202%20%5Ctimes%202%20-%207%20%3D%20%20-%2014)
The equation of tangent at:
x=2
![y - h(2) = h'(2)(x - 2) \\ y - 26 = - 14(x - 2) \\ y - 26 = - 14x + 28 \\ y = - 14x + 28 + 26 \\ y = - 14x + 54](https://tex.z-dn.net/?f=y%20-%20h%282%29%20%3D%20h%27%282%29%28x%20-%202%29%20%5C%5C%20y%20-%2026%20%3D%20%20-%2014%28x%20-%202%29%20%5C%5C%20y%20-%2026%20%3D%20%20-%2014x%20%2B%2028%20%5C%5C%20y%20%3D%20%20-%2014x%20%2B%2028%20%2B%2026%20%5C%5C%20y%20%3D%20%20-%2014x%20%2B%2054)
![\boxed{h(x) = - {x}^{2} - 7x + 44 \to \: y = - 14x + 54}](https://tex.z-dn.net/?f=%20%5Cboxed%7Bh%28x%29%20%3D%20%20-%20%20%7Bx%7D%5E%7B2%7D%20%20%20-%20%207x%20%20%20%2B%2044%20%5Cto%20%5C%3A%20y%20%3D%20%20%20-%2014x%20%20%20%2B%2054%7D)
<span class="sg-text sg-text--link sg-text--bold sg-text--link-disabled sg-text--blue-dark">
pdf
</span>
Answer:
y = 3.5
Step-by-step explanation:
First, you do 14/4 which = 3.5
Hope this helps!
Answer:
1). x° + 100° + 120° + 100° = 360°
2). x = 40
Step-by-step explanation:
Formula for the sum of the interior angles of a polygon = ![(n-2)\times 180](https://tex.z-dn.net/?f=%28n-2%29%5Ctimes%20180)
Where n = number of sides of the polygon
From the picture attached,
For n = 4,
Sum of the interior angles of the given polygon = (4 - 2) × 180°
= 360°
Therefore, equation for the sum of interior angles will be,
x° + 100° + 120° + 100° = 360°
x° + 320° = 360°
x° = 360° - 320°
x° = 40°
Answer:
x = 8, and y = 12
Step-by-step explanation:
There are 2 variables, so you need 2 equations to form a system of equations in two variables.
The upper left triangle has all angle measures given: 100, 2x + y, 5x + y. We know that the sum of the measures of the angles of a triangle is 180.
First equation:
100 + 2x + y + 5x + y = 180
Simplify:
7x + 2y = 80 (First equation)
Now we see that the upper and lower sides are parallel, so alternate interior angles are congruent. The angles measuring 2x + y and 5x - y are alternate interior angles and are congruent.
Second equation:
2x + y = 5x - y
Simplify:
3x - 2y = 0 (Second equation)
Now we use the first equation and the second equation as a system of simultaneous equations to solve for x and y.
7x + 2y = 80
3x - 2y = 0
Solve the second equation for 2x.
3x = 2y
Now replace 2y in the first equation with 3x.
7x + 3x = 80
10x = 80
x = 8
Replace x with 8 in the second equation.
3(8) - 2x = 0
24 = 2x
x = 12
Answer: x = 8, and y = 12