First solve the congruence [tex]13y\equiv1\pmod{330}[/tex]. Euclid's algorithm shows
330 = 25 * 13 + 5
13 = 2 * 5 + 3
5 = 1 * 3 + 2
3 = 1 * 2 + 1
=> 1 = 127 * 13 - 5 * 330
=> 127 * 13 = 1 mod 330
so that [tex]y=127[/tex] is the inverse of 13 modulo 330. Then in the original congruence, multiplying both sides by 127 twice gives
[tex]127^2\cdot13^2x\equiv127^2\cdot5^2\pmod{330}\implies x\equiv127^2\cdot5^2\equiv403,225\equiv295\pmod{330}[/tex]
Then any integer of the form [tex]x=295+330n[/tex] is a solution to the congruence, where [tex]n[/tex] is any integer.
Final answer:
Using the Chinese Remainder Theorem, the final solution to the congruence is x = 9 + 330k, where k is an integer.
Explanation:
To solve the congruence 169x ≡ 25 (mod 330), we can use the Chinese Remainder Theorem. First, we factor 330 into its prime factors: 330 = 2 × 3 × 5 × 11. Next, we solve the congruences 169x ≡ 25 (mod 2), 169x ≡ 25 (mod 3), 169x ≡ 25 (mod 5), and 169x ≡ 25 (mod 11) separately.
For the congruence 169x ≡ 25 (mod 2), since 169 ≢ 1 (mod 2), we can ignore it. For the congruence 169x ≡ 25 (mod 3), we can rewrite it as x ≡ 1 (mod 3). For the congruence 169x ≡ 25 (mod 5), we can rewrite it as 4x ≡ 0 (mod 5), and since gcd(5, 4) = 1, we can divide both sides by 4 to get x ≡ 0 (mod 5). Lastly, for the congruence 169x ≡ 25 (mod 11), we can rewrite it as 5x ≡ 3 (mod 11), and solve for x which will give us x ≡ 9 (mod 11).
Using the Chinese Remainder Theorem, we can combine these solutions to get x ≡ 9 (mod 330). Therefore, the solution to the congruence is x = 9 + 330k, where k is an integer.
Consider a drug testing company that provides a test for marijuana usage. Among 300 tested subjects, results from 30 subjects were wrong (either a false positive or a false negative). Use a 0.01 significance level to test the claim that less than 10 percent of the test results are wrong.
Answer:
static value come under the rejection value because it is less than critical value
Step-by-step explanation:
Given data
test = 300
wrong test = 30
significance level = 0.01
claim for wrong = 10 %
to find out
test the claim that less than 10 percent of the test results are wrong
solution
we take test claim null hypo thesis = 10 % = 0.10
and and alternate hypo thesis < 10% i.e. <0.10
and we know proportion of sample is = result/ test
sample proportion = 30/300 = 0.10
so the statistics of this test will be = sample proportion - hypothesis / [tex]\sqrt{hyro(1-hypo)/test}[/tex]
so statistics of this test = 0.10 - 0.10 / [tex]\sqrt{0.10(1-0.10)/300}[/tex]
so statistics of this test = 0
and α = tail area critical value for Z (0.01) = 2.33
so here static value come under the rejection value because it is less than critical value
To test the claim that less than 10 percent of the test results are wrong, we'll use a hypothesis test with a 0.01 significance level. The calculated test statistic z will determine whether to reject or fail to reject the null hypothesis.
Explanation:To test the claim that less than 10 percent of the test results are wrong, we'll use a hypothesis test with a 0.01 significance level. Let p represent the proportion of wrong test results. The null hypothesis is that p is greater than or equal to 0.10, and the alternative hypothesis is that p is less than 0.10.
We'll calculate the test statistic z = (x - np) / sqrt(np(1-p)), where x is the number of wrong test results and n is the total number of tested subjects. Using the given information, x = 30 and n = 300.
Using a z-table or calculator, we can find the z-score corresponding to a significance level of 0.01. If the calculated test statistic z is less than the z-score from the table, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
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Suppose a preliminary screening is given to prospective student athletes at a university to determine whether they would qualify for a scholarship. The scores are approximately normal with a mean of 85 and a standard deviation of 20. If the range of possible scores is 0 to 100, what percentage of students has a score less than 85?
Answer:
80
Step-by-step explanation:
Answer:
50% of students has a score less than 85
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 85, \sigma = 20[/tex]
What percentage of students has a score less than 85?
This is the pvalue of Z when X = 85. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{85 - 85}{20}[/tex]
[tex]Z = 0[/tex]
[tex]Z = 0[/tex] has a pvalue of 0.5
50% of students has a score less than 85
A total of 391 tickets were sold for the school play. They were either adult tickets or student tickets. There were 59 fewer student tickets sold than adult tickets. How many adult tickets were sold?
Answer:
There are 225 adult tickets.
Step-by-step explanation:
Let the number of adult tickets but represented by A while the number of student tickets be represented by S.
We are told the total is 391, so A+S=391.
There are 59 fewer student tickets (S) sold than adult tickets (T).
So there are 59 more adult tickets which means, A=59+S.
We are going to solve the system:
A+S=391
A=59+S
--------------------------------------------------
We are going to input the second equation into the first:
A+S=391 with A=59+S
(59+S)+S=391
59+S+S=391
59+2S=391
Subtract 59 on both sides:
2S=391-59
2S=332
Divide both sides by 2:
S=332/2
S=166
So A=59+S given S=166 means A=59+166=225.
There are 225 adult tickets while there is 166 student tickets.
Let's say the adult tickets would be X and the student tickets would be
X - 59.
Let's add them up. X + X - 59 = 391
2x = 450
x = 225
If there are 225 student tickets, we know that there are 63 fewer adult tickets so we take 225 and subtract 59 and we will get an answer of 166.
225 adult tickets were sold.
find the integral of ((x^(2)-1))/(x^(2)+3x) using integration by partial fractions
Answer:
[tex]x+\frac{-1}{3} \ln|x|+\frac{-8}{3}\ln|x+3|+C[/tex]
Step-by-step explanation:
To use partial fractions, I'm going to first do long division because the degree of the top is more than or equal to that of the bottom.
After I have that the degree of the bottom is more than the degree of the top, I will factor my bottom to figure out what kinds of partial fractions I'm going to have.
Let's begin with the long division:
The bottom goes outside.
The top goes inside.
1
-----------------
x^2+3x| x^2 -1
-(x^2+3x)
-----------------------------
-3x -1
We can not going any further since the divisor is more in degree than the left over part.
So we have so for that the integrand given equals:
[tex]\frac{x^2-1}{x^2+3x}=1+\frac{-3x-1}{x^2+3x}[/tex]
The 1 will of course not need partial fraction.
So we know our answer is x + something + C
Since the derivative of (x+c)=(1+0)=1.
Let's focus now on:
[tex]\frac{-3x-1}{x^2+3x}[/tex]
The bottom is not too bad too factor because it is binomial quadratic containing terms with a common factor of x:
[tex]x^2+3x=x(x+3)[/tex]
Since both factors our linear and there are two factors, then we will have two partial fractions where the numerators are both constants.
So we are looking to make this true:
[tex]\frac{-3x-1}{x^2+3x}=\frac{A}{x}+\frac{B}{x+3}[/tex]
Some people like to combine the fractions on the left and then regroup the terms and then compare coefficients.
Some people also prefer a method called heaviside method.
So I'm actually going to do this last way and I will explain it as I got.
We are going to clear the fractions by multiplying both sides by [tex]x(x+3)[/tex] giving me:
[tex]-3x-1=A(x+3)+Bx[/tex]
I know x+3 will be 0 when x=-3 so entering in -3 for x gives:
[tex]-3(-3)-1=A(-3+3)+B(-3)[/tex]
[tex]9-1=A(0)-3B[/tex]
[tex]8=-3B[/tex]
Divide both sides by -3:
[tex]\frac{8}{-3}=B[/tex]
[tex]B=\frac{-8}{3}[/tex]/
Now let's find A. If I replace x with 0 then Bx becomes 0 giving me:
[tex]-3(0)-1=A(0+3)+B(0)[/tex]
[tex]-1=A(3)+0[/tex]
[tex]-1=3A[/tex]
Divide both sides by 3:
[tex]\frac{1}{-3}=A[/tex]
[tex]\frac{-1}{3}=A[/tex]
Okay let me also who you other method of just comparing coefficients.
[tex]-3x-1=A(x+3)+Bx[/tex]
Distribute on the right:
[tex]-3x-1=Ax+3A+Bx[/tex]
Regroup terms on right so like terms are together:
[tex]-3x-1=(A+B)x+3A[/tex]
Now if this is to be true then we need:
-3=A+B and -1=3A
The second equation can be solved by dividing both sides by 3 giving us:
-3=A+B and -1/3=A
Now we are going to plug that second equation into the first:
-3=A+B with A=-1/3
-3=(-1/3)+B
Add 1/3 on both sides:
-3+(-1/3)=B
-8/3=B
So either way you should get the same A and B if no mistake is made of course.
So this is the integral we are looking at now (I'm going to go ahead and include the 1 from earlier):
[tex]\int (1+\frac{\frac{-1}{3}}{x}+\frac{\frac{-8}{3}}{x+3})dx[/tex]
[tex]x+\frac{-1}{3} \ln|x|+\frac{-8}{3}\ln|x+3|+C[/tex]
Why I did choose natural log for both of those 2 terms' antiderivatives?
Because they are a constant over a linear expression. Luckily both of those linear expressions had a leading coefficient of 1.
Also recall the derivative of ln(x) is (x)'/x=1/x and
the derivative of ln(x+3) is (x+3)'/(x+3)=(1+0)/(x+3)=1/(x+3).
Let's check our answer:
To do that we need to differentiate what we have for the integral and see if we wind up with the integrand.
[tex]\frac{d}{dx}(x+\frac{-1}{3} \ln|x|+\frac{-8}{3}\ln|x+3|+C)[/tex]
[tex]1+\frac{-1}{3} \frac{1}{x}+\frac{-8}{3}\frac{1}{x+3}+0[/tex]
We are going to find a common denominator which would be the least common multiple of the denominators which is x(x+3):
[tex]\frac{x(x+3)+\frac{-1}{3}(x+3)+\frac{-8}{3}(x)}{x(x+3)}[/tex]
Now distribute property:
[tex]\frac{x^2+3x+\frac{-1}{3}x-1+\frac{-8}{3}x}{x^2+3x}[/tex]
Combine like terms:
[tex]\frac{x^2+3x+\frac{-9}{3}x-1}{x^2+3x}[/tex]
[tex]\frac{x^2+3x-3x-1}{x^2+3x}[/tex]
[tex]\frac{x^2-1}{x^2+3x}[/tex]
So that is the same integrand we started with so our answer has been confirmed.
Write down the converse and contrapositive. If 2 divides x, then x is an even number. Hint call this (p?q)
Answer:
The converse and contrapositive of the given conditional statement are :
Converse : "If x is an even number, then 2 divides x".
Contrapositive : "If x is not an even number, then 2 does not divide x."
Step-by-step explanation: We are given to write the converse and contrapositive of the following conditional statement :
"If 2 divides x, then x is an even number."
Let us consider that
p : 2 divides x
and
q : x is an even number.
We know that
the CONVERSE of a conditional statement p ⇒ q is written as q ⇒ p.
So, the converse of the given statement is
"If x is an even number, then 2 divides x".
The CONTRAPOSITIVE of the conditional statement p ⇒ q is written as ~q ⇒ ~p (where ~p stands for the negation of p).
So, the contrapositive of the given statement is
"If x is not an even number, then 2 does not divide x."
Thus, the converse and contrapositive of the given statement are :
Converse : "If x is an even number, then 2 divides x".
Contrapositive : "If x is not an even number, then 2 does not divide x."
A realtor sold a home for $341,100 The commission was 4% of the sale price; however, the realtor receives only 60% of the commission while 40% of the commission remains with his broker. Find the amount of commission received by the realtor.
Answer:
The relator recived $8,186.4
Step-by-step explanation:
What you have to do is find the 60% of the commission. Commission is 4% of $341,100 (100%)
First do a cross multiplication to find 4% of $341,100
100% ___ $341,100
4%______x:
[tex]x=(4*341,100)/100=13,644[/tex]
So, the 4% of $341,100 is $13,466
Now you have to find the 60% of $13,466
100% ___ $13,466
60%______x:
[tex]x=(60*13,466)/100=8,186.4[/tex]
The answer is: $8,186.4
The marketing manager for a newspaper has commissioned a study of the advertisements in the classified section. The results for the Wednesday edition showed that 208 are help dash wanted ads, 552 are real estate ads, and 331 are other ads. a. If the newspaper plans to select an ad at random each week to be published free, what is the probability that the ad for a specific week will be a help dash wanted ad
Answer:
[tex]\frac{208}{1091}[/tex]
Step-by-step explanation:
To find probability of x, we need to find the number of x divided by total number.
Here,
Total = 208 + 552 + 331 = 1091
Number of help dash wanted ad = 208
Hence, the probability that the ad for a specific week is a help dash wanted ad = 208/1091
Find a counter example to show that the following statement is false:
For all nonzero real numbers a, b, c, d, a/b + c/d = a+c/b+d
A. a=3, b=5,c=-3,d=5
B. a=0, b=4,c=0,d=9
C. a=-2, b=1,c=2,d=1
D.a=1, b=2,c=1,d=2
Answer:
Option D. is the answer.
Step-by-step explanation:
The given statement is [tex]\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}[/tex]
Now we have to find a counter example from the given options that the statement is False.
A. a = 3, b = 5, c = -3, d = 5
[tex]\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}[/tex]=[tex]\frac{3}{5}+\frac{-3}{5}=\frac{3-3}{5+5}[/tex]
0 = 0
So the given statement is true.
B. a = 0, b = 4, c = 0, d= 9
[tex]\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}[/tex]
0 + 0 = 0
So for this example the given statement is true.
C. a = -2, b = 1, c = 2, d = 1
[tex]\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}[/tex]
[tex]\frac{-2}{1}+\frac{2}{1}=\frac{-2+2}{1+1}[/tex]
0 = 0
Statement is true for these values.
D. a = 1, b = 2, c = 1, d = 2
[tex]\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}[/tex]
[tex]\frac{1}{2}+\frac{1}{2}=\frac{1+1}{2+2}[/tex]
[tex]1=\frac{1}{2}[/tex]
Therefore, for these values of a, b, c and d, the given statement is False.
Option D. is the answer.
To find a counter example for the given statement, we need to test the options provided. However, all the options satisfy the equation, indicating that the statement holds true for all nonzero real numbers.
Explanation:To find a counter example for the given statement, we need to find values for a, b, c, and d that make the equation a/b + c/d = a+c/b+d false. Let's consider the options given:
Option A: a=3, b=5, c=-3, d=5Option B: a=0, b=4, c=0, d=9Option C: a=-2, b=1, c=2, d=1Option D: a=1, b=2, c=1, d=2We can see that for all the options, the equation holds true. Therefore, there is no counter example and the statement is true for all nonzero real numbers.
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I need the answer to this question.
1) Divide 1197 hours 48 minutes by 53.
Need hours and minutes
Round to the nearest hundredth as necessary.
1197 hours, 48 minutes
is equal to
1197 + 48/60 hours
Dividing by 53 gives
1197/53 + 48/3180 hours
Since
1197 = 22*53 + 31
we get
22 + 31/53 + 48/3180 hours
22 + (1860 + 48)/3180 hours
22 + 3/5 hours
22 + 36/60 hours
22 hours, 36 minutes
Use the definition to find an expression for the area under the curve y = x3 from 0 to 1 as a limit. lim n→∞ n i = 1 R (b) The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a). 13 + 23 + 33 + + n3 = n(n + 1) 2 2
The summand (R?) is missing, but we can always come up with another one.
Divide the interval [0, 1] into [tex]n[/tex] subintervals of equal length [tex]\dfrac{1-0}n=\dfrac1n[/tex]:
[tex][0,1]=\left[0,\dfrac1n\right]\cup\left[\dfrac1n,\dfrac2n\right]\cup\cdots\cup\left[1-\dfrac1n,1\right][/tex]
Let's consider a left-endpoint sum, so that we take values of [tex]f(\ell_i)={\ell_i}^3[/tex] where [tex]\ell_i[/tex] is given by the sequence
[tex]\ell_i=\dfrac{i-1}n[/tex]
with [tex]1\le i\le n[/tex]. Then the definite integral is equal to the Riemann sum
[tex]\displaystyle\int_0^1x^3\,\mathrm dx=\lim_{n\to\infty}\sum_{i=1}^n\left(\frac{i-1}n\right)^3\frac{1-0}n[/tex]
[tex]=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=1}^n(i-1)^3[/tex]
[tex]=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=0}^{n-1}i^3[/tex]
[tex]=\displaystyle\lim_{n\to\infty}\frac{n^2(n-1)^2}{4n^4}=\boxed{\frac14}[/tex]
The limit expression for the area under the curve y = x³ as n approaches infinity is; ¹/₄
What is the integral limit?The given definition is area A of the region S that lies under the graph of the continuous function which is the limit of the sum of the areas of approximating rectangles.
The expression for the area under the curve y = x³ from 0 to 1 as a limit is;
[tex]\lim_{n \to \infty} \Sigma^{n} _{i = 1} (\frac{i}{n})^{3} * \frac{1}{n} }[/tex]
From the expression above, when we factor out 1/n⁴, we will get;
[tex]\lim_{n \to \infty} \frac{1}{n^{4} } \Sigma^{n} _{i = 1} \frac{i}{n} }[/tex]
This is further broken down to get;
[tex]\lim_{n \to \infty} \frac{1}{n^{4} } (\frac{n(n + 1)}{2} )^{2} } }[/tex]
This will be simplified to;
[tex]\lim_{n \to \infty} \frac{1}{n^{4} } \frac{(n^{4} + 2n^{3} + n^{2}) }{2}[/tex]
This would be simplified to;
[tex]\lim_{n \to \infty} \frac{1}{4} + \frac{1}{2n} + \frac{1}{4n^{2} }[/tex]
At limit of n approaches ∞, we have;
Limit = ¹/₄
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Let A {a, b, c, d}. Find the power set of A, PA
[tex]A=\{a,b,c,d\}\\|A|=4\\\mathcal{P}(A)=2^4=16[/tex]
A researcher wishes to estimate the proportion of adults who have high-speed Internet access. What size sample should be obtained if she wishes the estimate to be within 0.05 with 95% confidence if (a) she uses a previous estimate of 0.32? (b) she does not use any prior estimates?
Answer: a) 8359 b) 384
Step-by-step explanation:
Given : Significance level : [tex]\alpha=1-0.95=0.05[/tex]
Critical value : [tex]z_{\alpha/2}}=\pm1.96[/tex]
Margin of error : [tex]E=0.01[/tex]
a) If previous estimate of proportion : [tex]p=0.32[/tex]
Formula to calculate the sample size needed for interval estimate of population proportion :-
[tex]n=p(1-p)(\frac{z_{\alpha/2}}{E})^2[/tex]
[tex]\Rightarrow\ n=0.32(1-0.32)(\frac{1.96}{0.01})^2=8359.3216\approx 8359[/tex]
Hence, the required sample size would be 8359 .
b) If she does not use any prior estimate , then the formula to calculate sample size will be :-
[tex]n=0.25\times(\frac{z_{\alpha/2}}{E})^2\\\\\Rightarrow\ n=0.25\times(\frac{1.96}{0.05})^2=384.16\approx384[/tex]
Hence, the required sample size would be 384 .
a rain gutter is 24 feet long, 4 inches in height, 3 inches at base and 6 inches at top. how many gallons of water will it hold?
Answer:1.8701
Step-by-step explanation:
Area (A) of trapezium as marked is =[tex]\frac{1}{2}[/tex][tex]\left ( sum\ of\ parallel\ sides\right )\times height[/tex]
A=[tex]\frac{1}{2}[/tex][tex]\left ( 3+6\right )\times 4[/tex]
A=18[tex]in^2[/tex]
Now for Volume we have to multiply by length because area is same across the length.
volume(v)=[tex]A\times length[/tex]
Volume(v)=[tex]18\times 24[/tex]
Volume(v)=432 [tex]in^3[/tex]
Volume(v)=1.8701 gallon
Calculate the percent of customer returns for a week with the following figures: gross sales, $7,500; returns, $95 on Monday, $19 on Tuesday, $50 on Wednesday, $140 on Thursday, $160 on Friday, and $80 on Saturday.
Answer:
The percentage of customer returns is 7%.
Step-by-step explanation:
Given,
Total gross sales = $ 7,500
Returns are,
$95 on Monday, $19 on Tuesday, $50 on Wednesday, $140 on Thursday, $160 on Friday, and $80 on Saturday,
So, the total returns = $ 95 + $ 50 + $ 140 + $ 160 + $ 80 = $ 525,
Hence, the percent of customer returns for the week = [tex]\frac{\text{Total returns}}{\text{Total gross sales}}\times 100[/tex]
[tex]=\frac{525}{7500}\times 100[/tex]
[tex]=\frac{52500}{7500}[/tex]
[tex]=7\%[/tex]
Find the size of each of 6 payments made at the end of each year into a 6% rate sinking fund which produces $82000 at the end of 6 years. The payment size is $? rounded to the nearest cent
Answer:
at the end of 1st year we pay $ 11756
at the end of 2nd year we pay $ 24217
at the end of 3rd year we pay $ 37426
at the end of 4th year we pay $ 51427
at the end of 5th year we pay $ 66269
at the end of 6th year we pay $ 82001
Step-by-step explanation:
Given data
rate ( i ) = 6%
Future payment = $82000
no of time period ( n ) = 6
to find out
size of all of 6 payments
solution
we know future payment formula i.e.
future payment = payment per period ( [tex](1 + rate)^{n}[/tex] - 1 ) / rate
put all these value and get payment per period
payment per period = future payment × rate / ( [tex](1 + rate)^{n}[/tex] - 1 ) / rate
payment per period = 82000 × 0.06 / ( [tex](1 + 0.06)^{6}[/tex] - 1 ) / rate
payment per period = 82000 × 0.06 / 0.4185
payment per period = $ 11756.27
at the end of 1st year we pay $ 11756
and at the end of 2nd year we pay $ 11756 × ( 1 + 0.06) + 11756
and at the end of 2nd year we pay $ 24217
and at the end of 3rd year we pay $ 24217 × ( 1 + 0.06) + 11756
and at the end of 3rd year we pay $ 37426
and at the end of 4th year we pay $ 37426 × ( 1 + 0.06) + 11756
and at the end of 4th year we pay $ 51427
and at the end of 5th year we pay $ 51427 × ( 1 + 0.06) + 11756
and at the end of 5th year we pay $ 66269
and at the end of 6th year we pay $ 66269 × ( 1 + 0.06) + 11756
and at the end of 6th year we pay $ 82001
a pizza parlor offers a choice of 12 different toppings. how many 3-topping pizzas are possible? (no double orders of toppings are allowed)
4 pizzas with 3 toppings each
only 12 toppings and 3 toppings per pizza
You have to do 12/3=4
given the drawing what would the value of x need to be in order for it to be true
Answer:
13
Step-by-step explanation:
For m||n to be true, we would need to have those angles equal, or 7x-21 = 4x+18.
Solve this equation, and get 3x=39 or x=13.
value of x = 13.
What are the properties of line and angle?Equal angles are those that are vertically opposed, such as a = d and b = c. Adjacent angles add up to 180 degrees, as in the cases where a + b and a + c. Angles that correspond to one another are equal, such as a=e, b=f, c=g, and d=h. Interior angles combine to make 180 degrees, such as c plus e or d plus f. Equal alternate angles are c = f and d = e.
Given a figure, one line is intersecting the two parallel lines known as m and n. since both given angles are equal alternate angles hence both angles would be the same.
Hence,
4x + 18 = 7x - 21
x =13
Therefore, in the given drawing the value of x needs to be 13.
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Which of the formulas below could be a polynomial with all of the following properties: its only zeros are x = -6, -2, 2, it has y-intercept y = 4, and its long-run behavior is y rightarrow - infinity as x plusminus infinity? Select every formula that has all of these properties. A. y = -4/144 (x + 6)^2 (x + 2)(x - 2) B. y = -4/192 (x + 6)(x + 2)^4 (x - 2) C. y = -4x (x + 6)(x + 2)(x - 2) D. y = -4/24 (x + 6)(x + 2)(x - 2) E. y = -4/48 (x + 6) (x + 2)^2 (x - 2) F. y = -4/48 (x + 6)(x + 2)(x - 2)^2 G. y = 4/48 (x + 6)(x + 2) (x - 2)^2
Answer:
A, B, and E
if I read your functions right.
Step-by-step explanation:
It's zeros are x=-6,-2, and 2.
This means we want the factors (x+6) and (x+2) and (x-2) in the numerator.
It has a y-intercept of 4. This means we want to get 4 when we plug in 0 for x.
And it's long-run behavior is y approaches - infinity as x approaches either infinity. This means the degree will be even and the coefficient of the leading term needs to be negative.
So let's see which functions qualify:
A) The degree is 4 because when you do x^2*x*x you get x^4.
The leading coefficient is -4/144 which is negative.
We do have the factors (x+6), (x+2), and (x-2).
What do we get when plug in 0 for x:
[tex]\frac{-4}{144}(0+6)^2(0+2)(0-2)[/tex]
Put into calculator: 4
A works!
B) The degree is 6 because when you do x*x^4*x=x^6.
The leading coefficient is -4/192 which is negative.
We do have factors (x+6), (x+2), and (x-2).
What do we get when we plug in 0 for x:
[tex]\frac{-4}{192}(0+6)(0+2)^4(x-2)[/tex]
Put into calculator: 4
B works!
C) The degree is 4 because when you do x*x*x*x=x^4.
The leading coefficient is -4 which is negative.
Oops! It has a zero at 0 because of that factor of (x) between -4 and (x+6).
So C doesn't work.
D) The degree is 3 because x*x*x=x^3.
We needed an even degree.
D doesn't work.
E) The degree is 4 because x*x^2*x=x^4.
The leading coefficient is -4/48 which is negative.
It does have the factors (x+6), (x+2), and (x-2).
What do we get when we plug in 0 for x:
[tex]\frac{-4}{48}(0+6)(0+2)^2(0-2)[/tex]
Put into calculator: 4
So E does work.
F) The degree is 4 because x*x*x^2=x^4.
The leading coefficient is -4/48.
It does have factors (x+6), (x+2), and (x-2).
What do we get when we plug in 0 for x:
[tex]\frac{-4}{48}(0+6)(0+2)(0-2)^2[/tex]
Put into calculator: -4
So F doesn't work.
G. I'm not going to go any further. The leading coefficient is 4/48 and that is not negative.
So G doesn't work.
Answer:
D. [tex]y=-\frac{4}{24} (x+6)(x+2)(x-2)[/tex]Step-by-step explanation:
Notice that we have 3 zeros, which means there are only 3 roots, which are -6, -2 and 2, this indicates that our expression must be cubic with the binomials (x+6), (x+2) and (x-2).
We this analysis, possible choices are C and D.
Now, according to the problem, it has y-intercept at y = 4, so let's evaluate each expression for x = 0.
C.[tex]y=-4x(x+6)(x+2)(x-2)\\y=-4(0)(0+6)(0+2)(0-2)\\y=0[/tex]
D.[tex]y=-\frac{4}{24} (x+6)(x+2)(x-2)\\y=-\frac{4}{24}(0+6)(0+2)(0-2)\\y=-\frac{4}{24}(-24)\\ y=4[/tex]
Therefore, choice D is the right expression because it has all given characteristics.
a gross of baseball cards contains 144 packs of cards. each pack contains 8 baseball cards how many cards do you have in all?
912 cards in all
If there are 114 packs of cards and 8 in each you multiply 114 by 8 and get 912.
To find the total number of baseball cards, multiply the number of packs (144) by the number of cards in each pack (8), resulting in a total of 1,152 cards.
Explanation:If a gross of baseball cards contains 144 packs of cards and each pack contains 8 baseball cards, the total number of cards would be calculated by multiplying the number of packs by the number of cards in each pack.
To get the total number of baseball cards.
144 packs x 8 cards/pack = 1,152 cards
Hence, a gross of baseball cards contains 1,152 cards in total.
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An automatic filling machine in a factory fills bottles of ketchup with a mean of 17.6 oz and a standard deviation of 0.05 oz with a distribution that can be well modeled by a Normal model. What is the probability that a bottle of ketchup contains less than 17.5 oz?
The probability that a bottle of ketchup contains less than 17.5 oz, given a normal distribution with a mean of 17.6 oz and a standard deviation of 0.05 oz, is approximately 2.28%.
Explanation:The question is asking for the probability that a bottle of ketchup contains less than 17.5 oz, given that the distribution of ketchup amounts in the bottle is normally distributed with a mean of 17.6 oz and a standard deviation of 0.05 oz. In terms of a normal distribution chart, we are looking for the area under the curve to the left of 17.5 oz.
To find this, we need to first convert our 17.5 oz to a standard score (or z-score). The z-score represents how many standard deviations away from the mean a data point is. It is calculated using the formula:
z = (X - μ) / σ
Where X is the data point (17.5 oz), μ is the mean (17.6 oz), and σ is the standard deviation (0.05 oz). Using this formula, we find z = -2.
Now, to find the probability, we can refer to a standard normal distribution table. The area under the curve to the left of z = -2 (which represents a data point of 17.5 oz) is approximately 0.0228, or 2.28%. Therefore, the probability that a bottle of ketchup contains less than 17.5 oz is about 2.28%.
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Let p and q denote the following statements. pis "The chair is broken." qis "The stove is hot." Write the compound statement. The chair is broken or the stove is not hot, in symbolic form. The chair is broken or the stove is not hot. The symbolic form is a
Final answer:
The statement 'The chair is broken or the stove is not hot' translates to 'p v ~q' in symbolic form, where 'p' represents 'The chair is broken,' and 'q' represents 'The stove is hot.' The 'v' symbolizes 'or,' and '~' indicates negation.
Explanation:
The question requests the compound statement 'The chair is broken or the stove is not hot' be written in symbolic form using propositional logic. Given the statements, p is 'The chair is broken,' and q is 'The stove is hot,' we can translate the request into symbolic form. The use of 'or' in propositional logic is represented by the symbol 'v' (wedge), and the negation of a statement is represented by the symbol '~' (tilde). Therefore, 'The stove is not hot' can be represented as '~q'.
To combine these statements into the requested compound statement, we use:
p v ~q
This reads as 'p or not q', which translates back into the original statement 'The chair is broken or the stove is not hot.' This allows us to see how logical operators like disjunction ('or') and negation ('not') work together to construct compound statements in propositional logic.
PLEASE HELP
I am confused on these equations
Answer:
See below.
Step-by-step explanation:
I'll show you step by step how to do the synthetic division.
You are dividing polynomial -6x^4 + 22x^3 + 3x^2 + 15x + 20 by x - 4.
From the polynomial, you only need the coefficients in descending order of degree as they are written.
-6 22 3 15 20
In synthetic division, you divide by x - b. You are dividing by x - 4, so b = 4.
The 4 is written to the left of the coefficients of the polynomial.
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4 | -6 22 3 15 20
___________________
This is what the setup looks like. You can see it in the problem you were given.
The first step is to just copy the leftmost coefficient straight down to below the line.
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4 | -6 22 3 15 20
___________________
-6
Now multiply b, which is 4, by -6 and write it above and to the right.
___________________
4 | -6 22 3 15 20
-24
___________________
-6
Add 22 and -24 and write it next to -6.
___________________
4 | -6 22 3 15 20
-24
___________________
-6 -2
Multiply 4 by -2 and write it above and to the right. Add vertically.
___________________
4 | -6 22 3 15 20
-24 -8
___________________
-6 -2 -5
Multiply 4 by -5 and write it above and to the right. Add vertically.
___________________
4 | -6 22 3 15 20
-24 -8 -10
___________________
-6 -2 -5 -5
Multiply 4 by -5 and write it above and to the right. Add vertically.
___________________
4 | -6 22 3 15 20
-24 -8 -10 -20
___________________
-6 -2 -5 -5 0
The 4 numbers in the second line are the coefficients of the quotient.
The quotient is 1 degree less than the original polynomial.
The quotient is: -24x^3 - 8x^2 - 10x - 20
The last number on the third line is the remainder. Since here the reminder is zero, that means that this division has no remainder. The remainder is 0/(x - 4)
Fill in the boxes with:
-24, -8, -10, -20
-6, -2, -5, -5, 0
-24x^3 - 8x^2 - 10x - 20
0
Write the rational expression in lowest terms. x^2-9x+8/x^2 + x-3; x^2-9x+8/x^2+2x-3 (Simplify your answer.)
Answer:
[tex]\frac{x-8}{x+3}[/tex]
Step-by-step explanation:
Given expression,
[tex]\frac{x^2-9x+8}{x^2+2x-3}[/tex]
[tex]=\frac{x^2-(8+1)x+8}{x^2+(3-1)x-3}[/tex]
[tex]=\frac{x^2-8x-x+8}{x^2+3x-x-3}[/tex]
[tex]=\frac{x(x-8)-1(x-8)}{x(x+3)-1(x+3)}[/tex]
[tex]=\frac{(x-1)(x-8)}{(x-1)(x+3)}[/tex]
[tex]=\frac{x-8}{x+3}[/tex]
Since, further simplification is not possible,
Hence, the given rational expression in lowest terms is,
[tex]\frac{x-8}{x+3}[/tex]
4 Permutations 1. Let a "small block cipher be a function f which maps 8-bit plaintexts m E (0,1)8 to 8-bit ciphertexts c E (0,1)8. However, function f must be 1-1 or otherwise it would be impossible to invert a block cipher, i.e. compute (using the block cipher key) the plaintext f(c) given ciphertext c. How many "small" block ciphers are there? students if each student must have a computer and computers cannot be shared? capital letters but with no repeated characters? 2. There are 10 computers and 5 students. In how many ways can computers be assigned to 3. Recall problem 1. How many 8-character passwords are there made of either lower-case or . How many 10-digit decimal strings are there in which there is no repeated digits? 5. How many 10-digit decimal strings are there in which there is no repeated digits and where 5" occurs before "6"? (By "before" I mean "anywhere before" and not "mmediately before", e.g. string (5,0, 1, 2, 3, 4, 7, 8, 9, 6) is included.) 6. How many 10-digit decimal strings are there in which there is no repeated digits and where "5" and "6" are next to each other?
The population of a town with a 2016 population of 66,000 grows at a rate of 2.5% per year a. Find the rate constant k and use it to devise an exponential growth function that fits the given data b. In what year will the population reach 176.000? Book a. Find the rate constant k k= (Type an exact answer) tents ccess Library Resources
Final answer:
The rate constant, k, can be found using the formula k = ln(1 + r), where r is the growth rate. In this case, the growth rate is 2.5%. The rate constant is approximately 0.0253. The exponential growth function is P(t) = 66,000 * e^(0.0253t). To find in what year the population will reach 176,000, we solve the equation 176,000 = 66,000 * e^(0.0253t) and find that it will take approximately 42 years.
Explanation:
To find the rate constant, we can use the formula:
k = ln(1 + r)
where k is the rate constant and r is the growth rate as a decimal.
In this case, the growth rate is 2.5%, which is equivalent to 0.025 as a decimal.
Using the formula, we have:
k = ln(1 + 0.025) = ln(1.025) ≈ 0.0253
Therefore, the rate constant k is approximately 0.0253.
To devise an exponential growth function, we can use the formula:
P(t) = P0 * ekt
where P(t) is the population at time t, P0 is the initial population, k is the rate constant, and t is the time in years.
In this case, the initial population P0 is 66,000 and we already found that the rate constant k is 0.0253.
So, the exponential growth function is:
P(t) = 66,000 * e0.0253t
To find in what year the population will reach 176,000, we can set up the following equation:
176,000 = 66,000 * e0.0253t
Divide both sides by 66,000:
176,000 / 66,000 = e0.0253t
Simplify:
2.6667 = e0.0253t
To solve for t, we can take the natural logarithm of both sides:
ln(2.6667) = 0.0253t
Divide both sides by 0.0253:
t = ln(2.6667) / 0.0253 ≈ 41.71
Therefore, the population will reach 176,000 in approximately 41.71 years, which can be rounded to 42 years.
if a full oil barrel holds 53 3/4 gallons of oil, and 10 3/4 gallons are used every week, how many weeks will the oil last?
The value is 5 weeks
To find out how many weeks the oil will last with a specific usage per week, divide the total oil amount by the weekly usage, resulting in 5 weeks.
The oil barrel holds 53 3/4 gallons of oil, and 10 3/4 gallons are used every week.
To determine how many weeks the oil will last, you need to divide the total amount of oil by the weekly usage.
53 3/4 gallons ÷ 10 3/4 gallons per week = 5 weeks
In the diagram, how many pairs of vertical angles are shown?
I see 4 pairs of vertical angles and they are:
7/9
8/10
11/13
12/14
Answer:
4 pairs
Step-by-step explanation:
vertical angles are opposite angles formed by two intersecting lines.
please thank if you learned from this answer.
Solve the inequality and graph the solution. -12+ 5(9p+3)-36p <6p-12
Answer:
Given inequality is,
-12 + 5(9p+3) - 36p < 6p - 12
By distributive property,
-12 + 45p + 15 - 36p < 6p - 12
Combine like terms,
3 + 9p < 6p - 12
Additive property of equality,
15 + 9p < 6p
Subtraction property of equality,
9p < 6p - 15
3p < -15
⇒ p < -5
That is, solution of the given inequality is (-∞, 5)
Robin has 15% of his gross pay directly into his mutual fund account each month. if $630 is depisited each month. what is his gross pay?
Answer:
Robin's gross pay=$50,400
Step-by-step explanation:
We are given that Robin has 15% of his gross pay directly into his mutual fund account each month.
If Robin deposited each month =$630
We have to find the gross pay
Let Robin's gross pay =x
15 % of gross pay= 15% of x=[tex]\frac{15}{100}\times x=\frac{15}{100}x[/tex]
We know that 12 month in a year
[tex] \frac{15}{100}x=630\times 12[/tex]
[tex] 15x=630\times 100\times 12[/tex]
Using multiplication property of equality
[tex] x=\frac{756000}{15}[/tex]
Using division property of equality
[tex]x=50400[/tex]
Hence, Robin's gross pay=$50,400
The lengths of pregnancies are normally distributed with a mean of 267 days and a standard deviation of 15 days. a. Find the probability of a pregnancy lasting 309 days or longer. b. If the length of pregnancy is in the lowest 4%, then the baby is premature. Find the length that separates premature babies from those who are not premature.
Answer:
We have a normal distribution with a mean of 267 days and a standard deviation of 15 days. To solve this proble we're going to need the help of a calculator.
a. The probability of a pregnancy lasting 309 days or longer is:
P(z>309) = 0.0026 or 0.26%
b. The lowest 4% is separeted by the 240.74 days. The probability of pregnancy lasting 240.74 days is 4%.
Answer:
a) 0.26% probability of a pregnancy lasting 309 days or longer.
b) A pregnancy length of 241 days separates premature babies from those who are not premature.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 267, \sigma = 15[/tex]
a. Find the probability of a pregnancy lasting 309 days or longer.
This is 1 subtracted by the pvalue of Z when X = 309. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{309 - 267}{15}[/tex]
[tex]Z = 2.8[/tex]
[tex]Z = 2.8[/tex] has a pvalue of 0.9974
So there is a 1-0.9974 = 0.0026 = 0.26% probability of a pregnancy lasting 309 days or longer.
b. If the length of pregnancy is in the lowest 4%, then the baby is premature. Find the length that separates premature babies from those who are not premature.
This is the value of X when Z has a pvalue of 0.04. So X when Z = -1.75
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.75 = \frac{X - 267}{15}[/tex]
[tex]X - 267 = -1.75*15[/tex]
[tex]X = 240.75[/tex]
A pregnancy length of 241 days separates premature babies from those who are not premature.