Answer:
She must consider 3507 components to be 90% sure of knowing the mean will be within ± 0.1 mm.
Step-by-step explanation:
We are given that an engineer wishes to determine the width of a particular electronic component. If she knows that the standard deviation is 3.6 mm.
And she considers to be 90% sure of knowing the mean will be within ±0.1 mm.
As we know that the margin of error is given by the following formula;
The margin of error = [tex]Z_(_\frac{\alpha}{2}_) \times \frac{\sigma}{\sqrt{n} }[/tex]
Here, [tex]\sigma[/tex] = standard deviation = 3.6 mm
n = sample size of components
[tex]\alpha[/tex] = level of significance = 1 - 0.90 = 0.10 or 10%
[tex]\frac{\alpha}{2} = \frac{0.10}{2}[/tex] = 0.05 or 5%
Now, the critical value of z at a 5% level of significance in the z table is given to us as 1.645.
So, the margin of error = [tex]Z_(_\frac{\alpha}{2}_) \times \frac{\sigma}{\sqrt{n} }[/tex]
0.1 mm = [tex]1.645 \times \frac{3.6}{\sqrt{n} }[/tex]
[tex]\sqrt{n} = \frac{3.6\times 1.645}{0.1 }[/tex]
[tex]\sqrt{n}[/tex] = 59.22
n = [tex]59.22^{2}[/tex] = 3507.0084 ≈ 3507.
Hence, she must consider 3507 components to be 90% sure of knowing the mean will be within ± 0.1 mm.
generate a continuous and differentiable function f(x) with the following properties: f(x) is decreasing at x=−5 f(x) has a local minimum at x=−3 f(x) has a local maximum at x=3
Answer:
see details in graph and below
Step-by-step explanation:
There are many ways to generate the function.
We'll generate a function whose first derivative f'(x) satisfies the required conditions, say, a quadratic.
1. f(x) has a local minimum at x = -3, and
2. a local maximum at x = 3
Therefore f'(x) has to cross the x-axis at x = -3 and x=+3.
Furthermore, f'(x) must be increasing at x=-3 and decreasing at x=+3.
f'(x) = -x^2+9
will satisfy the above conditions.
Finally f(x) must be decreasing at x= -5, which implies that f'(-5) must be negative.
Check: f'(-5) = -(-5)^2+9 = -25+9 = -16 < 0 so ok.
f(x) can then be obtained by integrating f'(x) :
f(x) = integral of -x^2+9 = -x^3/3 + 9x = 9x - x^3/3
A graph of f(x) is attached, and is found to satisfy all three conditions.
A function is differentiable at [tex]x = a[/tex], if the function is continuous at [tex]x = a[/tex]. The function that satisfy the given properties is [tex]f(x) = 9x - \frac{x^3}{3} + 3[/tex]
Given that:
The function decreases at [tex]x = -5[/tex] means that: [tex]f(-5) < 0[/tex]
The local minimum at [tex]x = -3[/tex] and local maximum at [tex]x = 3[/tex] means that:
[tex]x = -3[/tex] or [tex]x = 3[/tex]
Equate both equations to 0
[tex]x + 3 = 0[/tex] or [tex]3 - x = 0[/tex]
Multiply both equations to give y'
[tex]y' = (3 - x) \times (x + 3)[/tex]
Open bracket
[tex]y' = 3x + 9 - x^2 - 3x[/tex]
Collect like terms
[tex]y' = 3x - 3x+ 9 - x^2[/tex]
[tex]y' = 9 - x^2[/tex]
Integrate y'
[tex]y = \frac{9x^{0+1}}{0+1} - \frac{x^{2+1}}{2+1} + c[/tex]
[tex]y = \frac{9x^1}{1} - \frac{x^3}{3} + c[/tex]
[tex]y = 9x - \frac{x^3}{3} + c[/tex]
Express as a function
[tex]f(x) = 9x - \frac{x^3}{3} + c[/tex]
[tex]f(-5) < 0[/tex] implies that:
[tex]9\times -5 - \frac{(-5)^3}{3} + c < 0[/tex]
[tex]-45 - \frac{-125}{3} + c < 0[/tex]
[tex]-45 + \frac{125}{3} + c < 0[/tex]
Take LCM
[tex]\frac{-135 + 125}{3} + c < 0[/tex]
[tex]-\frac{10}{3} + c < 0[/tex]
Collect like terms
[tex]c < \frac{10}{3}[/tex]
[tex]c <3.33[/tex]
We can then assume the value of c to be
[tex]c=3[/tex] or any other value less than 3.33
Substitute [tex]c=3[/tex] in [tex]f(x) = 9x - \frac{x^3}{3} + c[/tex]
[tex]f(x) = 9x - \frac{x^3}{3} + 3[/tex]
See attachment for the function of f(x)
Read more about continuous and differentiable function at:
https://brainly.com/question/19590547
Which transformation was applied to Figure 1 in order to arrive at Figure 2? Geometry A
Answer:
(B) Reflection in the x-axis
Step-by-step explanation:
We can see that these triangles have the exact same x-coordinates, however their y coordinates are opposite each other. This means that if we wanted to get one of the triangles to the other, we’d have to reflect over the x-axis
(by default, if the x values are the same and y are opposite, reflect across x axis. If y values are the same and x is opposite, reflect over y. it’s sort of like opposites.)
Hope this helped!
What will be the effect on the graph of y = Ixl if x is replaced with -x?
Answer:
If x is replaced with -x the graph will stay the same because the absolute value makes 2 values so a negative number and a positive one.
Step-by-step explanation:
Go search it up on desmos.
Find the product of
the sum of
3/5 and 1%
and
Answer:
3/500
Step-by-step explanation:
3/5 x 1%
=> 3/5 x 1/100
=> 3/500
Hope it helps you
Simplify using calculator.. I'm not sure if i am putting it in the calculator right
You would type in
32^(6/5)
Or you could type in
32^(1.2)
since 6/5 = 1.2
Either way, the final result is 64
Identify the inverse function of f(x) = VX - 2 + 3.
Answer:
[tex]\huge\boxed{f^{-1}(x) = (x-3)^2+2}[/tex]
Step-by-step explanation:
[tex]f(x) = \sqrt{x-2} + 3[/tex]
Replace y = f(x)
[tex]y = \sqrt{x-2} + 3[/tex]
Exchange x and y
[tex]x = \sqrt{y-2}+3[/tex]
Solve for y
[tex]x = \sqrt{y-2}+3[/tex]
Subtracting both sides by 3
[tex]x - 3 = \sqrt{y-2}[/tex]
Taking square on both sides
[tex](x-3)^2 = y -2[/tex]
Adding 2 to both sides
[tex]y = (x-3)^2+2[/tex]
Substitute y = [tex]f^{-1}(x)[/tex]
[tex]f^{-1}(x) = (x-3)^2+2[/tex]
Answer:
[tex] \boxed{ {f}^{ - 1} (x) = {(x - 3)}^{2} + 2}[/tex]Option D is the correct option
Step-by-step explanation:
[tex] \mathsf{f(x) = \sqrt{x - 2} + 3}[/tex]
Replace f(x) with y
[tex] \mathsf{y = \sqrt{x - 2} + 3}[/tex]
Interchange variables
[tex] \mathsf{x = \sqrt{y - 2} + 3}[/tex]
[tex] \mathsf{{(x - 3)}^{2} = {( \sqrt{y - 2)} }^{2} }[/tex]
[tex] \mathsf{ {(x - 3)}^{2} = y - 2}[/tex]
[tex] \mathsf{ y = {(x - 3)}^{2} + 2}[/tex]
Replace y with f ⁻¹( x )
[tex] \mathsf{ {f}^{ - 1} (x) = {(x - 3)}^{2} + 2}[/tex]
Hope I helped!
Best regards!
I will rate brainly if you answer this The number of weekly social media posts varies directly with the square root of the poster’s age and inversely with the cube root of the poster’s income. If a 16-year-old person who earns $8,000 makes 64 posts in a week, what is the value of k?
Answer:
[tex]\large \boxed{\sf \bf \ \ k=320 \ \ }[/tex]
Step-by-step explanation:
Hello,
The number of weekly social media posts varies directly with the square root of the poster’s age and inversely with the cube root of the poster’s income.
If a 16-year-old person who earns $8,000 makes 64 posts in a week, what is the value of k?
[tex]64=\dfrac{\sqrt{16}}{\sqrt[3]{8000}}\cdot k=\dfrac{4}{20}\cdot k=\dfrac{1}{5}\cdot k=0.2\cdot k\\\\k=64*5=320[/tex]
Hope this helps.
Do not hesitate if you need further explanation.
Thank you
An octagonal pyramid ... how many faces does it have, how many vertices and how many edges? A triangular prism ... how many faces does it have, how many vertices and how many edges? a triangular pyramid ... how many faces does it have, how many vertices and how many edges?
1: 8 faces and 9 with the base 9 vertices and 16 edges
2: 3 faces and 5 with the bases 6 vertices and 9 edges
3: 3 faces and 4 with the base 4 vertices and 6 edges
Hope this can help you.
1: 8 faces and 9 with the base 9 vertices and 16 edges
2: 3 faces and 5 with the bases 6 vertices and 9 edges
3: 3 faces and 4 with the base 4 vertices and 6 edges
Multiply the following complex numbers:
(7+2i)(2+3i)
Please don’t guess
Answer:
14 + 25l + 6l^2
Step-by-step explanation:
(7 + 2i) (2 + 3i)
=> 14 + 4l + 21l + 6l^2
=> 14 + 25l + 6l^2
This is the correct answer
The Tran family and the Green family each used their sprinklers last summer. The water output rate for the Tran family's sprinkler was 35L per hour. The water output rate for the Green family's sprinkler was 40L per hour. The families used their sprinklers for a combined total of 50 hours, resulting in a total water output of 1900L. How long was each sprinkler used?
Answer:
Tran family's sprinkler was used for 20 hours
Green's family's sprinkler was used for 30 hours
Step-by-step explanation:
Let the hours for which Tran family's sprinkler used is x hours
water output rate for the Tran family's sprinkler = 35L per hour
water output from Tran family's sprinkler in x hours = 35*x L = 35x
Let the hours for which Green family's sprinkler used is y hours
water output rate for the Green family's sprinkler = 40L per hour
water output from Green family's sprinkler in x hours = 40*y L = 40y
Given
The families used their sprinklers for a combined total of 50 hours
thus
x + y = 50 -------------------equation 1
y = 50-x
total water output of 1900L
35x+40y = 1900 -------------------equation 1
using y = 50-x in equation 2, we have
35x + 40(50-x) = 1900
35x + 2000 - 40x = 1900
=> -5x = 1900 - 2000 = -100
=> x = -100/-5 = 20
y = 50-20 = 30
Thus,
Tran family's sprinkler was used for 20 hours
Green's family's sprinkler was used for 30 hours
If f(x)=ax+b/x and f(1)=1 and f(2)=5, what is the value of A and B?
Answer:
[tex]\huge\boxed{a=9 ; b = -8}[/tex]
Step-by-step explanation:
[tex]f(x) = \frac{ax+b}{x}[/tex]
Putting x = 1
=> [tex]f(1) = \frac{a(1)+b}{1}[/tex]
Given that f(1) = 1
=> [tex]1 = a + b[/tex]
=> [tex]a+b = 1[/tex] -------------------(1)
Now,
Putting x = 2
=> [tex]f(2) = \frac{a(2)+b}{2}[/tex]
Given that f(2) = 5
=> [tex]5 = \frac{2a+b}{2}[/tex]
=> [tex]2a+b = 5*2[/tex]
=> [tex]2a+b = 10[/tex] ----------------(2)
Subtracting (2) from (1)
[tex]a+b-(2a+b) = 1-10\\a+b-2a-b = -9\\a-2a = -9\\-a = -9\\a = 9[/tex]
For b , Put a = 9 in equation (1)
[tex]9+b = 1\\Subtracting \ both \ sides \ by \ 9\\b = 1-9\\b = -8[/tex]
Need help with this problem ASAP, don’t need an explanation, just an answer
Answer:
x^3-10x^2+1/9
Step-by-step explanation:
For standard form you need to put the exponents in order. So x^3 is first, followed by -10x^2, and finally 1/9. Hope this helps!
find the value of each variable and the measure of each angle
Answer:
y = 90x = 302x° = 60°(y+x)° = 120°(y-x)° = 60°Step-by-step explanation:
Adjacent angles are supplementary, so ...
(y +x) +(y -x) = 180
2y = 180 . . . . . . . . . simplify
y = 90 . . . . . . . . . . . divide by 2
__
2x +(y +x) = 180
3x +90 = 180 . . . . substitute for y
x + 30 = 60 . . . . . . divide by 3
x = 30 . . . . . . . . . . subtract 30
__
With these values of x and y, the angle measures are ...
2x° = 2(30)° = 60°
(y+x)° = (90+30)° = 120°
(y-x)° = (90-30)° = 60°
Construct a polynomial function with the following properties: fifth degree, 4 is a zero of multiplicity 3, −2 is the only other zero, leading coefficient is 2.
Answer:
[tex]\Large \boxed{\sf \bf \ \ 2(x-4)^3(x+2)^2 \ \ }[/tex]
Step-by-step explanation:
Hello, please consider the following.
Construct a polynomial function with the following properties...
... fifth degree
It means that the polynomial can be written as below.
[tex]a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \ \text{ with }a_5\text{ different from 0}\\\\\text{ or } k(x-x_1)(x-x_2)(x-x_3)(x-x_4)(x-x_5) \\\\ \text{ with k different from 0 and } (x_i)_{1\leqi\leq 5 } \text { are the roots.}[/tex]
... 4 is a zero of multiplicity 3
We can write the polynomial as below.
[tex]k(x-4)(x-4)(x-4)(x-x_4)(x-x_5)=k(x-4)^3(x-x_4)(x-x_5)[/tex]
... −2 is the only other zero
Because this is the only other zero, we can deduce that -2 is a zero of multiplicity 2.
[tex]k(x-4)(x-4)(x-4)(x-x_4)(x-x_5)\\\\=k(x-4)^3(x-(-2))(x-(-2))\\\\=k(x-4)^3(x+2)^2[/tex]
... leading coefficient is 2.
Finally, it means that k = 2 and then the polynomial function is:
[tex]\large \boxed{2(x-4)^3(x+2)^2}[/tex]
Hope this helps.
Do not hesitate if you need further explanation.
Thank you
A nutrition laboratory tested 25 "reduced sodium" hotdogs of a certain brand, finding that the mean sodium content is 310 mg with a standard deviation of 36 mg.
Construct a 95% confidence interval for the mean sodium content of this brand of hot dog and interpret a 95% level of confidence. Show all work
Answer:
The 95% confidence interval is [tex]295.9 < \mu< 324.1[/tex]
A 95% level of confidence mean that there is 95% chance that the true population mean will be in this interval
Step-by-step explanation:
From the question we are told that
The sample size is [tex]n = 25[/tex]
The mean is [tex]\= x = 310 \ mg[/tex]
The standard deviation is [tex]\sigma = 36 \ mg[/tex]
Given that the confidence level is 95% then the level of significance is mathematically represented as
[tex]\alpha = 100 - 95[/tex]
=> [tex]\alpha = 5\%[/tex]
=> [tex]\alpha = 0.05[/tex]
Next we obtain the critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table , the value is
[tex]Z_{\frac{\alpha }{2} } =Z_{\frac{0.05 }{2} } = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }[/tex]
substituting values
[tex]E = 1.96 * \frac{36 }{\sqrt{25} }[/tex]
[tex]E = 14.1[/tex]
The 95% level of confidence interval is mathematically represented as
[tex]\= x - E < \mu<\ \= x - E[/tex]
substituting values
[tex]310- 14.1 < \mu< 310+ 14.1[/tex]
[tex]295.9 < \mu< 324.1[/tex]
The 95% level of confidence mean that there is 95% chance that the true population mean will be in this interval
The heights of North American women are nor-mally distributed with a mean of 64 inches and a standard deviation of 2 inches. a. b. c. What is the probability that a randomly selected woman is taller than 66 inches
Answer:
0.1587
Step-by-step explanation:
Given the following :
Mean (m) of distribution = 64 inches
Standard deviation (sd) of distribution = 2 inches
Probability that a randomly selected woman is taller than 66 inches
For a normal distribution :
Z - score = (x - mean) / standard deviation
Where x = 66
P(X > 66) = P( Z > (66 - 64) / 2)
P(X > 66) = P(Z > (2 /2)
P(X > 66) = P(Z > 1)
P(Z > 1) = 1 - P(Z ≤ 1)
P(Z ≤ 1) = 0.8413 ( from z distribution table)
1 - P(Z ≤ 1) = 1 - 0.8413
= 0.1587
i will rate you brainliest// What is the interquartile range (IQR) of {5.8, 8.5, 9.9, -0.8, -1.3, 2.3, 7.4, -1.9}?
Answer
arrange the element in increasing order
-1.9, -1.3, -0.8, 2.3, 5.8, 7.4, 8.5, 9.9
interquatile = Q3 - Q1
[tex] = \frac{7.4 + 8.5}{2} - \frac{ - 1.3 - 0.8}{2} [/tex]
[tex] = 7.95 + 1.05[/tex]
[tex] = 9[/tex]
Answer:
9.0
Step-by-step explanation:
i took the quiz
The Airline Passenger Association studied the relationship between the number of passengers on a particular flight and the cost of the flight. It seems logical that more passengers on the flight will result in more weight and more luggage, which in turn will result in higher fuel costs. For a sample of 21 flights, the correlation between the number of passengers and total fuel cost was 0.668.
(1)
State the decision rule for 0.10 significance level: H0: Ï â‰¤ 0; H1: Ï > 0 (Round your answer to 3 decimal places.)
Reject H0 if t >
(2)
Compute the value of the test statistic. (Round your answer to 3 decimal places.)
Value of the test statistic
Answer:
Decision Rule: To reject the null hypothesis if t > 1.328
t = 3.913
Step-by-step explanation:
The summary of the given statistics include:
sample size n = 21
the correlation between the number of passengers and total fuel cost r = 0.668
(1) We are tasked to state the decision rule for 0.10 significance level
The degree of freedom df = n - 1
degree of freedom df = 21 - 1
degree of freedom df = 19
The null and the alternative hypothesis can be computed as:
[tex]H_o : \rho < 0\\ \\ Ha : \rho > 0[/tex]
The critical value for [tex]t_{\alpha, df}[/tex] is [tex]t_{010, 19}[/tex] = 1.328
Decision Rule: To reject the null hypothesis if t > 1.328
The test statistics can be computed as follows by using the formula for t-test for Pearson Correlation:
[tex]t = r*\sqrt{ \dfrac{(n-2)}{(1-r^2)}[/tex]
[tex]t = 0.668*\sqrt{ \dfrac{(21-2)}{(1-0.668^2)}[/tex]
[tex]t = 0.668*\sqrt{ \dfrac{(19)}{(1-0.446224)}[/tex]
[tex]t = 0.668*\sqrt{ \dfrac{(19)}{(0.553776)}[/tex]
[tex]t = 0.668*5.858[/tex]
t = 3.913144
t = 3.913 to 3 decimal places
An octagonal pyramid ... how many faces does it have, how many vertices and how many edges? A triangular prism ... how many faces does it have, how many vertices and how many edges? a triangular pyramid ... how many faces does it have, how many vertices and how many edges?
1: 8 faces and 9 with the base 9 vertices and 16 edges
2: 3 faces and 5 with the bases 6 vertices and 9 edges
3: 3 faces and 4 with the base 4 vertices and 6 edges
Hope this can help you.
1: 8 faces and 9 with the base 9 vertices and 16 edges
2: 3 faces and 5 with the bases 6 vertices and 9 edges
3: 3 faces and 4 with the base 4 vertices and 6 edges
On a coordinate plane, a line has points (negative 2, negative 4) and (4, 2). Point P is at (0, 4). Which points lie on the line that passes through point P and is parallel to the given line? Select three options. (–4, 2) (–1, 3) (–2, 2) (4, 2) (–5, –1)
Answer:
the correct options are:
(–1, 3), (–2, 2) and (–5, –1)
Step-by-step explanation:
Given that a line passes through two points
A(-2, -4) and B(4, 2)
Another point P(0, 4)
To find:
Which points lie on the line that passes through P and is parallel to line AB ?
Solution:
First of all, let us the find the equation of the line which is parallel to AB and passes through point P.
Parallel lines have the same slope.
Slope of a line is given as:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\dfrac{2-(-4)}{4-(-2)} = 1[/tex]
Now, using slope intercept form ([tex]y = mx+c[/tex]) of a line, we can write the equation of line parallel to AB:
[tex]y =(1)x+c \Rightarrow y = x+c[/tex]
Now, putting the point P(0,4) to find c:
[tex]4 = 0 +c \Rightarrow c = 4[/tex]
So, the equation is [tex]\bold{y=x+4}[/tex]
So, the coordinates given in the options which have value of y coordinate equal to 4 greater than x coordinate will be true.
So, the correct options are:
(–1, 3), (–2, 2) and (–5, –1)
Answer:
b,c,e
Step-by-step explanation:
I got it right on edge
8.What side of the road will you see speed, yield, and guide signs on ?
Answer:
we see it in our left side of the road
Can someone help me?
Answer:
7w
Step-by-step explanation:
Solve the following system of linear equations {2x-7y=10 {5x -6y=2
2x-7y=10 = [tex]\frac{2}{7}[/tex]
5x -6y=2 = [tex]\frac{5}{6}[/tex]
Compute using long division: 1,234÷68
Answer:
Quotient = 18
Remainder = 10
Step-by-step explanation:
1234/68
=> 68 x 1 = 68
=> 123 - 68 = 55
=> Take the 4 down
=> 554/68
=> 68 x 8 = 544
=> 554 - 544 = 10
So, the quotient = 18.
Remainder = 10
G={3,7,8,9} h={2,5,7,8} what is the intersection of the sets
Answer:
The answer is { 7 , 8 }Step-by-step explanation:
G = { 3 , 7 , 8 , 9 }
H = { 2 , 5 , 7 , 8 }
The intersection of any two or more sets are the members that occur in both sets.
To find the intersection of G and H look for the members that occur in both sets
From the question , the members that occur in both G and H are 7 and 8
So the intersection of the sets is
{ 7 , 8 }Hope this helps you
In the last 10 years, the population of Indonesia has grown at a rate of 1.12% per year to 258,316,051. If this rate continues, what will be the population in 10 more years? Round your answer to the nearest whole number.
Answer:
Final population after 10 years
= 288911718
Step-by-step explanation:
Present population p = 258,316,051
Rate of growth R%= 1.12%
Number of years t= 10 years
Number of times calculated n = 10
Final population A
= P(1+r/n)^(nt)
A= 258,316,051(1+0.0112/10)^(10*10)
A= 258,316,051(1+0.00112)^(100)
A= 258,316,051(1.00112)^100
A= 258,316,051(1.118442762)
A= 288911717.6
Approximately A= 288911718
Final population after 10 years
= 288911718
In the given figure, if POQ is a straight line then find ∠POT. please help !!!!!!
Answer:
∠POT = 78°
Step-by-step explanation:
If POQ is straight then
x + 18° + 50° + x + 24° = 180° add like terms
2x + 92° = 180°
2x = 180° - 92°
2x = 88° and x = 44 If we say SOT is a straight line then
∠POT + 50° + x + 18° = 180°
∠POT + 102° = 180°
∠POT = 78°
If 2 x 2 + 13 x − 7 = 0 , then x could equal which of the following?
Hi there! :)
Answer:
x = 1/2 or -7.
Step-by-step explanation:
(I'm assuming the expression is 2x² + 13x - 7 = 0)
Factor the equation to solve for the possible values of "x":
2x² + 13x - 7 = 0
When factored, we get:
(2x - 1) ( x + 7) = 0
Use the Zero-Product property to solve for the roots:
2x - 1 = 0
2x = 1
x = 1/2.
-----------
x + 7 = 0
x = -7.
Therefore, possible values of x are x = -1/2, 7.
Answer:
x = 1/2 x=-7
Step-by-step explanation:
2 x^2 + 13 x − 7 = 0
Factor
(2x-1)(x+7)=0
Using the zero product property
2x-1 =0 x+7=0
2x=1 x =-7
x = 1/2 x=-7
Can somebody explain how trigonometric form polar equations are divided/multiplied?
Answer:
Attachment 1 : Option C
Attachment 2 : Option A
Step-by-step explanation:
( 1 ) Expressing the product of z1 and z2 would be as follows,
[tex]14\left[\cos \left(\frac{\pi \:}{5}\right)+i\sin \left(\frac{\pi \:\:}{5}\right)\right]\cdot \:2\sqrt{2}\left[\cos \left(\frac{3\pi \:}{2}\right)+i\sin \left(\frac{3\pi \:\:}{2}\right)\right][/tex]
Now to solve such problems, you will need to know what cos(π / 5) is, sin(π / 5) etc. If you don't know their exact value, I would recommend you use a calculator,
cos(π / 5) = [tex]\frac{\sqrt{5}+1}{4}[/tex],
sin(π / 5) = [tex]\frac{\sqrt{2}\sqrt{5-\sqrt{5}}}{4}[/tex]
cos(3π / 2) = 0,
sin(3π / 2) = - 1
Let's substitute those values in our expression,
[tex]14\left[\frac{\sqrt{5}+1}{4}+i\frac{\sqrt{2}\sqrt{5-\sqrt{5}}}{4}\right]\cdot \:2\sqrt{2}\left[0-i\right][/tex]
And now simplify the expression,
[tex]14\sqrt{5-\sqrt{5}}+i\left(-7\sqrt{10}-7\sqrt{2}\right)[/tex]
The exact value of [tex]14\sqrt{5-\sqrt{5}}[/tex] = [tex]23.27510\dots[/tex] and [tex](-7\sqrt{10}-7\sqrt{2}\right))[/tex] = [tex]-32.03543\dots[/tex] Therefore we have the expression [tex]23.27510 - 32.03543i[/tex], which is close to option c. As you can see they approximated the solution.
( 2 ) Here we will apply the following trivial identities,
cos(π / 3) = [tex]\frac{1}{2}[/tex],
sin(π / 3) = [tex]\frac{\sqrt{3}}{2}[/tex],
cos(- π / 6) = [tex]\frac{\sqrt{3}}{2}[/tex],
sin(- π / 6) = [tex]-\frac{1}{2}[/tex]
Substitute into the following expression, representing the quotient of the given values of z1 and z2,
[tex]15\left[cos\left(\frac{\pi \:}{3}\right)+isin\left(\frac{\pi \:\:}{3}\right)\right] \div \:3\sqrt{2}\left[cos\left(\frac{-\pi \:}{6}\right)+isin\left(\frac{-\pi \:\:}{6}\right)\right][/tex] ⇒
[tex]15\left[\frac{1}{2}+\frac{\sqrt{3}}{2}\right]\div \:3\sqrt{2}\left[\frac{\sqrt{3}}{2}+-\frac{1}{2}\right][/tex]
The simplified expression will be the following,
[tex]i\frac{5\sqrt{2}}{2}[/tex] or in other words [tex]\frac{5\sqrt{2}}{2}i[/tex] or [tex]\frac{5i\sqrt{2}}{2}[/tex]
The solution will be option a, as you can see.
1. (a) Find the probability that a 90% free-throw shooter makes 10 consecutive free-throws, assuming that individual shots are independent.
Answer:
[tex]Probability = 0.35[/tex]
Step-by-step explanation:
Given
Probability of success free throw = 90%
Number of throw = 10
Required
Determine the probability of 10 consecutive free throws
Let p represents the given probability
[tex]p = 90\%[/tex]
Convert to decimal
[tex]p = 0.9[/tex]
Let n represents the number of throw
[tex]n = 10[/tex]
Provided that each throw is independent;
The probability of n consecutive free throw is
[tex]p^n[/tex]
Substitute 0.9 for p and 10 for n
[tex]Probability = 0.9^{10}[/tex]
[tex]Probability = 0.3486784401[/tex]
[tex]Probability = 0.35[/tex] (Approximated)
