We are asked to prove by the method of mathematical induction that:
3n(n+1) is divisible by 6 for all positive integers.
for n=1 we have:[tex]3n(n+1)=3\times 1(1+1)\\\\i.e.\\\\3n(n+1)=3\times 2\\\\i.e.\\\\3n(n+1)=6[/tex]
which is divisible by 6.
Hence, the result is true for n=1
Let the result is true for n=ki.e. 3k(k+1) is divisible by 6.
Now we prove that the result is true for n=k+1Let n=k+1
then
[tex]3n(n+1)=3(k+1)\times (k+1+1)\\\\i.e.\\\\3n(n+1)=3(k+1)(k+2)\\\\i.e.\\\\3n(n+1)=(3k+3)(k+2)\\\\i.e.\\\\3n(n+1)=3k(k+2)+3(k+2)\\\\i.e.\\\\3n(n+1)=3k^2+6k+3k+6\\\\i.e.\\\\3n(n+1)=3k^2+3k+6k+6\\\\i.e.\\\\3n(n+1)=3k(k+1)+6(k+1)[/tex]
Since, the first term:
[tex]3k(k+1)[/tex] is divisible by 6.
( As the result is true for n=k)
and the second term [tex]6(k+1)[/tex] is also divisible by 6.
Hence, the sum:
[tex]3k(k+1)+6(k+1)[/tex] is divisible by 6.
Hence, the result is true for n=k+1
Hence, we may say that the result is true for all n where n belongs to positive integers.
To prove by induction that the expression 3n(n+1) is divisible by 6, we start with the base case of n=1, which is divisible by 6, and then show that if it holds for an integer k, it also holds for k+1. By factoring and using the induction hypothesis, we demonstrate the expression's divisibility by 6 for all positive integers.
Proof by Induction of Divisibility by 6
To prove by induction that 3n(n+1) is divisible by 6 for all positive integers, we follow two steps: the base case and the inductive step.
Base Case
Let's check for n=1:
Inductive Step
Assume the statement holds for a positive integer k, so 3k(k+1) is divisible by 6.
Now, we must show that 3(k+1)((k+1)+1) = 3(k+1)(k+2) is also divisible by 6.
Factoring out the common term we get:
Notice that (k+1) is an integer, hence 3(k+1) is divisible by 3. If k is even, then k+1 is odd, so 3(k+1) is still divisible by 3 but not necessarily by 6. However, if k is odd, k+1 is even and 3(k+1) is divisible by 6. Since the divisible by 3 part is always true, and the divisible by an additional factor of 2 part is true every other time, the sum 6m + 3(k+1) is divisible by 6 regardless of whether k is odd or even. This is because adding a multiple of 6 to either another multiple of 6 or a multiple of 3 always results in a multiple of 6.
Therefore, 3n(n + 1) is divisible by 6 for all positive integers n.
Find and simplify the expression if
Answer:
[tex] f ( 2 x ) = 4 x ^ 2 - 8 [/tex]
Step-by-step explanation:
We are given the following expression and we are to simplify the given function:
[tex] f ( x ) = x ^ 2 - 8 [/tex]
Applying the function [tex]f(2x)[/tex] on [tex] f ( x ) = x ^ 2 - 8 [/tex] to get:
[tex] f ( 2 x ) = ( 2 x ) ^ 2 - 8 [/tex]
[tex] f ( 2 x ) = 4 x ^ 2 - 8 [/tex]
-1 mod 19 please show work a =dq+r
Answer:
The given expression can be written as -1=19(-1)+18.
Step-by-step explanation:
According to quotient remainder theorem:
For any integer a and a positive integer d, there exist unique integers q and r such that
[tex]a=d\times q+r[/tex]
It can also written as
[tex]a\text{mod }d=r[/tex]
The given expression is
[tex]-1\text{mod }19[/tex]
Here a=-1 and d=19.
[tex]\frac{-1}{19}=19(-1)+18[/tex]
[tex]-1\text{mod }19=18[/tex]
If -1 is divides by 19, then the remainder is 18. The value of r is 18.
Therefore the given expression can be written as -1=19(-1)+18.
-1 mod 19 is equal to 18.
Finding -1 mod 19
To determine -1 mod 19, we use the formula a = dq + r, where a = -1, d = 19, and q and r are the quotient and remainder, respectively.
First, note that any number mod 19 will yield a remainder between 0 and 18.
Since -1 is a negative number, we rewrite it in terms of 19: -1 = -1 + 19k for some integer k.
To find a positive equivalent, we choose k such that -1 + 19k is positive and falls within the range of 0 to 18. When k = 1, we get -1 + 19(1) = 18.
Therefore, -1 mod 19 is equal to 18.
This means that if you divide -1 by 19, the remainder that falls within the range of 0 to 18 is 18.
Use the method of reduction of order to find a second solution to t^2y' + 3ty' – 3y = 0, t> 0 Given yı(t) = t y2(t) = Preview Give your answer in simplest form (ie no coefficients)
Let [tex]y_2(t)=tv(t)[/tex]. Then
[tex]{y_2}'=tv'+v[/tex]
[tex]{y_2}''=tv''+2v'[/tex]
and substituting these into the ODE gives
[tex]t^2(tv''+2v')+3t(tv'+v)-3tv=0[/tex]
[tex]t^3v''+5t^2v'=0[/tex]
[tex]tv''+5v'=0[/tex]
Let [tex]u(t)=v'(t)[/tex], so that [tex]u'(t)=v''(t)[/tex]. Then the ODE is linear in [tex]u[/tex], with
[tex]tu'+5u=0[/tex]
Multiply both sides by [tex]t^4[/tex], so that the left side can be condensed as the derivative of a product:
[tex]t^5u'+5t^4u=(t^5u)'=0[/tex]
Integrating both sides and solving for [tex]u(t)[/tex] gives
[tex]t^5u=C\implies u=Ct^{-5}[/tex]
Integrate again to solve for [tex]v(t)[/tex]:
[tex]v=C_1t^{-6}+C_2[/tex]
and finally, solve for [tex]y_2(t)[/tex] by multiplying both sides by [tex]t[/tex]:
[tex]tv=y_2=C_1t^{-5}+C_2t[/tex]
[tex]y_1(t)=t[/tex] already accounts for the [tex]t[/tex] term in this solution, so the other independent solution is [tex]y_2(t)=t^{-5}[/tex].
Use the power series for 1 1−x to find a power series representation of f(x) = ln(1−x). What is the radius of convergence? (Note: you don’t need to use the ratio test here because we know the radius of convergence of the series P∞ n=0 x n .) (b) Use part (a) to find a power series for f(x) = x ln(1 − x). (c) By putting x = 1 2 in your result from part (a), express ln 2 as the sum of an infinite series
a. Recall that
[tex]\displaystyle\int\frac{\mathrm dx}{1-x}=-\ln|1-x|+C[/tex]
For [tex]|x|<1[/tex], we have
[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]
By integrating both sides, we get
[tex]\displaystyle-\ln(1-x)=C+\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}[/tex]
If [tex]x=0[/tex], then
[tex]\displaystyle-\ln1=C+\sum_{n=0}^\infty\frac{0^{n+1}}{n+1}\implies 0=C+0\implies C=0[/tex]
so that
[tex]\displaystyle\ln(1-x)=-\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}[/tex]
We can shift the index to simplify the sum slightly.
[tex]\displaystyle\ln(1-x)=-\sum_{n=1}^\infty\frac{x^n}n[/tex]
b. The power series for [tex]x\ln(1-x)[/tex] can be obtained simply by multiplying both sides of the series above by [tex]x[/tex].
[tex]\displaystyle x\ln(1-x)=-\sum_{n=1}^\infty\frac{x^{n+1}}n[/tex]
c. We have
[tex]\ln2=-\dfrac\ln12=-\ln\left(1-\dfrac12\right)[/tex]
[tex]\displaystyle\implies\ln2=\sum_{n=1}^\infty\frac1{n2^n}[/tex]
The power series of f(x) = ln(1 - x) is [tex]\rm -\sum^{\infty}_{n=1}\dfrac{x^{n} }{n}[/tex], the power series of xln(1 - x) is [tex]\rm -\sum^{\infty}_{n=1}\dfrac{x^{n+1} }{n}[/tex] and the value of ln(2) is [tex]\rm \sum^{\infty}_{n=0}\dfrac{1}{n2^n}[/tex].
Given :
f(x) = ln (1−x)
a) The integration of 1/(1 - x) is given by:
[tex]\rm \int \dfrac{1}{1-x}dx=-ln|1-x| + C[/tex]
When |x| >1 :
[tex]\dfrac{1}{1-x} = \sum^{\infty}_{n=0} x^n[/tex]
Now, integrate on both sides in the above equation.
[tex]\rm -ln(1-x) = C+\sum^{\infty}_{n=0}\dfrac{x^{n+1} }{n+1}[/tex] --- (1)
Now, at (x = 0) the above expression becomes:
[tex]\rm -ln(1-0) = C+\sum^{\infty}_{n=0}\dfrac{0^{n+1} }{n+1}[/tex]
By simplifying the above expression in order to get the value of C.
C = 0
Now, substitute the value of C in expression (1).
[tex]\rm ln(1-x) =-\sum^{\infty}_{n=0}\dfrac{x^{n+1} }{n+1}[/tex]
Now, by shifting the index the above expression becomes:
[tex]\rm ln(1-x) =-\sum^{\infty}_{n=1}\dfrac{x^{n} }{n}[/tex]
b) Now, multiply by 'x' in the above expression in order to get the power series of (x ln(1 - x)).
[tex]\rm xln(1-x) =-\sum^{\infty}_{n=1}\dfrac{x^{n+1} }{n}[/tex]
c) Now, substitute the value x = 1/2 in the expression (1).
[tex]\rm ln2 = \sum^{\infty}_{n=0}\dfrac{1}{n2^n}[/tex]
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Find and simplify each of the following for
Answer:
(A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
(B) f(x + h) - f(x) = 8xh + 4h² - 6h
(C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]
Step-by-step explanation:
* Lets explain how to solve the problem
- The function f(x) = 4x² - 6x + 6
- To find f(x + h) substitute x in the function by (x + h)
∵ f(x) = 4x² - 6x + 6
∴ f(x + h) = 4(x + h)² - 6(x + h) + 6
- Lets simplify 4(x + h)²
∵ (x + h)² = (x)(x) + 2(x)(h) + (h)(h) = x² + 2xh + h²
∴ 4(x + h)² = 4(x² + 2xh + h²) = 4x² + 8xh + 4h²
- Lets simplify 6(x + h)
∵ 6(x + h) = 6(x) + 6(h)
∴ 6(x + h) = 6x + 6h
∴ f(x + h) = 4x² + 8xh + 4h² - (6x + 6h) + 6
- Remember (-)(+) = (-)
∴ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
* (A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
- Lets find f(x + h) - f(x)
∵ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
∵ f(x) = 4x² - 6x + 6
∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - (4x² - 6x + 6)
- Remember (-)(-) = (+)
∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - 4x² + 6x - 6
- Simplify by adding the like terms
∴ f(x + h) - f(x) = (4x² - 4x²) + 8xh + 4h² + (- 6x + 6x) - 6h + (6 - 6)
∴ f(x + h) - f(x) = 8xh + 4h² - 6h
* (B) f(x + h) - f(x) = 8xh + 4h² - 6h
- Lets find [tex]\frac{f(x+h)-f(x)}{h}[/tex]
∵ f(x + h) - f(x) = 8xh + 4h² - 6h
∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh + 4h^{2}-6h}{h}[/tex]
- Simplify by separate the three terms
∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh}{h}+\frac{4h^{2} }{h}-\frac{6h}{h}[/tex]
∴ [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]
* (C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]
Using the Chinese Remainder Theorem, solve the congruence
x 15 (mod 42)
x 5 (mod 19)
19 and 42 are coprime, so we can use the CRT right away. Start with
[tex]x=19+42[/tex]
Taken mod 42, we're left with a remainder of 19. Notice that
[tex]19\cdot3\equiv57\equiv15\pmod{42}[/tex]
so we need to multiply the first term by 3 to get the remainder we want.
[tex]x=19\cdot3+42[/tex]
Next, taken mod 19, we're left with a remainder of 4. Notice that
[tex]42\cdot6\equiv252\equiv5\pmod{19}[/tex]
so we need to multiply the second term by 6.
Then by the CRT, we have
[tex]x\equiv19\cdot3+42\cdot6\equiv309\pmod{42\cdot19}\implies x\equiv309\pmod{798}[/tex]
so that any solution of the form [tex]x=798n+309[/tex] is a solution to this system.
Solve the Differential equation 2(y-4x^2) dx + x dy = 0
The solution of the given differential equation involves rearranging it into the standard form of a first-order linear differential equation, determining the integrating factor, and subsequently solving for the dependent variable y(x) via integration.
Explanation:To solve the given differential equation, we can rewrite it in the form of dy/dx = f(x, y). That gives us (2(y-4x^2))/x = dy/dx. The resulting equation is a first-order linear differential equation, which can be solved using an integrating factor.
Here, the standard form of the differential equation is dy/dx + P(x)y = Q(x). Comparing this with our equation, we find P(x) = -2/x and Q(x) = -8x. We know that μ(x) = exp(∫P(x) dx) is the integrating factor. On solving we get μ(x) = 1/x2. We then multiply through our differential equation by μ(x) and integrate both sides to solve for y(x).
These steps on how to solve the differential equation involve certain knowledge in differential equation theory, namely about first-order linear differential equations, integrating factors, and the process of integration.
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A professor has recorded exam grades for 10 students in his class, but one of the grades is no longer readable. If the mean score on the exam was 82 and the mean of the 9 readable scores is 84, what is the value of the unreadable score?
Answer:
64
Step-by-step explanation:
[tex]mean=\frac{sum\ of\ total\ number\ of\ score}{total\ number\ of\ students}[/tex]
we have given that mean of 9 students is 84
so total score of 9 students = mean×9
=84×9=756
and we have given mean score of exam is 82 and there is total 10 students so the total score of 10 students =10×82
=820
so the unreadable score = score of 10 students -score of 9 students =820-756=64
Thompson and Thompson is a steel bolts manufacturing company. Their current steel bolts have a mean diameter of 131 millimeters, and a standard deviation of 7 millimeters. If a random sample of 31 steel bolts is selected, what is the probability that the sample mean would differ from the population mean by more than 1.9 millimeters? Round your answer to four decimal places.
Answer: 0.1310
Step-by-step explanation:
Given : Mean : [tex]\mu = \text{131 millimeters}[/tex]
Standard deviation : [tex]\sigma = \text{7 millimeters}[/tex]
Sample size : [tex]n=31[/tex]
To find the probability that the sample mean would differ from the population mean by more than 1.9 millimeters i.e. less than 129.1 milliliters and less than 132.9 milliliters.
The formula for z-score :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x = 129.1 milliliters
[tex]z=\dfrac{129.1-131}{\dfrac{7}{\sqrt{31}}}\approx-1.51[/tex]
For x = 132.9 milliliters
[tex]z=\dfrac{132.9-131}{\dfrac{7}{\sqrt{31}}}\approx1.51[/tex]
The P-value= [tex]P(x<-1.51)+P(x>1.51)[/tex]
[tex]=2P(z>1.51)=2(1-P(z<1.15))\\=2(1-0.9344783)\\=0.1310434\approx0.1310[/tex]
Hence, the required probability = 0.1310
The probability that the sample mean would differ from the population mean by more than 1.9 millimeters is approximately 0.1744.
Explanation:To find the probability that the sample mean would differ from the population mean by more than 1.9 millimeters, we can use the Z-score formula. The Z-score is calculated by subtracting the population mean from the sample mean and dividing by the standard deviation divided by the square root of the sample size. Once we have the Z-score, we can use a Z-table or calculator to find the probability.
In this case, the Z-score is (1.9 - 0) / (7 / sqrt(31)) = 0.935. To find the probability of a Z-score greater than 0.935, we can look up the corresponding area under the normal distribution curve in a Z-table or use a calculator. The probability is approximately 0.1744.
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Additional Proofs: Prove each statement below using Proof by Contradiction. 1. The sum of any rational number and any irrational number is irrational. S 2. For all integers m, if m is even, then 3m+7 is odd I integer, then, 2a +3b is even 3. If a is any odd integer, and b is any even 4. Let A and B be sets from a universe U. If Bc A, then AC B
Answer:
See below.
Step-by-step explanation:
1. Suppose that the sum is rational then we can write:
a/b + i = c/d where i is irrational and by definition a/b and c/d are rational.
Rearranging:
i = c/d - a/b
Now the sum on the right is rational so 'irrational' = 'rational' which is a contradiction.
So the original supposition is false and the sum must be irrational.
2. Proof of For all integers m if m is even then 3m + 7 is odd:
If m is even then 3m is even.
Suppose 3m + 7 is even, then:
3m + 7 = 2p where p is an integer.
3m - 2p = -7
But 3m and 2p are both even so their result is even and -7 is odd.
Therefore the original supposition is false because it leads to a contradiction, so 3m + 7 is odd.
Given A = {3,4, 13, 15) and B = { 13, 15, 21, 25, 30, 37). Which of the following are true? I. AUB 13,4,21,25, 30, 37) III. ACB II. AnB-(13, 15) a) I and III only b) OII only e) Oll and III only d) I and II only e) OI, II, and III None of the above
The correct statements regarding the sets A and B is:
II.
[tex]A\bigcap B=\{13,15\}[/tex]
Step-by-step explanation:We are given set A as:
[tex]A=\{3,4,13,15\}[/tex]
and set B as:
[tex]B=\{13,15,21,25,30,37\}[/tex]
and the three statements are given by:
I.
[tex]A\bigcup B=\{12,4,21,25,30,37\}[/tex]
II.
[tex]A\bigcap B=\{13,15\}[/tex]
III.
[tex]A\subset B[/tex]
We know that the union of two sets is the collection of all the elements which belong either to A or to B.i.e. from the given sets we have:
[tex]A\bigcup B=\{3,4,13,15,21,25,30,37\}[/tex]
Also, the intersection of two sets is the collection of all the elements which belongs to both the set.i.e.
[tex]A\bigcap B=\{13,15\}[/tex]
Also, we know that A is a subset of B if:All the elements of A are contained in B.
But from the given sets we see that:
3,4 belongs to A but they do not belong to B.
Hence, A is not a subset of B.
Suppose that a department contains 9 men and 15 women. How many different committees of 6 members are possible if the committee must have strictly more women than men?
Answer: The required number of different possible committees is 81172.
Step-by-step explanation: Given that a department contains 9 men and 15 women.
We are to find the number of different committees of 6 members that are possible if the committee must have strictly more women than men.
Since we need committees of 6 members, so the possible combinations are
(4 women, 2 men), (5 women, 1 men) and (6 women).
Therefore, the number of different committees of 6 members is given by
[tex]n\\\\\\=^{15}C_4\times ^9C_2+^{15}C_5\times ^9C_1+^{15}C_6\\\\\\=\dfrac{15!}{4!(15-4)!}\times \dfrac{9!}{2!(9-2)!}+\dfrac{15!}{5!(15-5)!}\times \dfrac{9!}{1!(9-1)!}+\dfrac{15!}{6!(15-6)!}\\\\\\\\=\dfrac{15\times14\times13\times12\times11!}{4\times3\times2\times1\times11!}\times\dfrac{9\times8\times7!}{2\times1\times7!}+\dfrac{15\times14\times13\times12\times11\times10!}{5\times4\times3\times2\times1\times10!}\times\dfrac{9\times8!}{1\times8!}+\dfrac{15\times14\times13\times12\times11\times10\times9!}{6\times5\times4\times3\times2\times1\times9!}\\\\\\=1365\times36+3003\times9+5005\\\\=49140+27027+5005\\\\=81172.[/tex]
Thus, the required number of different possible committees of 6 members is 81172.
Consider the three points graphed. What is the value of y in the fourth point that will complete the quadrilateral as a rhombus?
The fourth point that will complete the quadrilateral is a rhombus is -3.
In a rhombus, two parallel sides must be equal.
What is a straight line graph?The graph follows a straight line equation shows a straight line graph.equation of a straight line is y=mx+cy represents vertical line y-axis.x represents the horizontal line x-axis. m is the slope of the lineslope(m)=tan∅=y axis/x axis.
c represents y-intercepts (it is the point at which the line cuts on the y-axis)Straight line graphs show a linear relationship between the x and y values.Calculation:-
1st coordinate- side=[tex]\sqrt{10}[/tex]
2nd coordinate- side= [tex]\sqrt{18}[/tex]
3rd coordinate must be [tex]\sqrt{10}[/tex]
so,y=-3
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Answer:
Consider the three points graphed. What is the value of y in the fourth point that will complete the quadrilateral as a rhombus?
(–1, -5 )
Step-by-step explanation:
Use the graph of the line to find the x-intercept, y-intercept, and slope. Write the slope-intercept form of the equation of the line.
1) x-intercept:
x-intercept is the point where the graph of the equation crosses the x-axis. From the given figure, we can see that the line is crossing the x-axis at -10. Thus the x-intercept is -9
2) y-intercept:
y-intercept is the point where the graph of the equation crosses the y-axis. From the given figure, we can see that the line is crossing the y-axis at -10. Thus the y-intercept is -9
3) Slope:
Slope of a line is calculated as:
[tex]slope=m=\frac{\text{Difference in y coordinates}}{\text{Difference in x coordinates}}[/tex]
For calculating the slope we can use both intercepts. x-intercept is ordered pair will be (-9, 0) and y-intercept will be (0, -9). So the slope of the line will be:
[tex]m=\frac{-9-0}{0-(-9)}=-1[/tex]
Therefore, the slope of the line is -1.
4) Slope intercept form of the line:
The slope intercept form of the line is represented as:
[tex]y=mx+c[/tex]
where,
m = slope of line = -1
c = y-intercept = -9
Using these values, the equation becomes:
[tex]y=-x- 9[/tex]
Answer:
x-intercept: [tex]-9[/tex].
y-intercept: [tex]-9[/tex].
Slope: [tex]-1[/tex]
Equation: [tex]y=-x-9[/tex]
Step-by-step explanation:
We have been given a graph of a line on coordinate plane. We are asked to find the x-intercept, y-intercept, and slope.
We know that x-intercept of a function is a point, where graph crosses x-axis.
Upon looking at our given graph, we can see that graph crosses x-axis at point [tex](-9,0)[/tex], therefore, x-intercept is [tex]-9[/tex].
We know that y-intercept of a function is a point, where graph crosses y-axis.
Upon looking at our given graph, we can see that graph crosses y-axis at point [tex](0,-9)[/tex], therefore, y-intercept is [tex]-9[/tex].
We have two points on the line. Let us find slope of line using points [tex](-9,0)[/tex] and [tex](0,-9)[/tex].
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\frac{-9-0}{0-(-9)}=\frac{-9}{0+9}=\frac{-9}{9}=-1[/tex]
Therefore, the slope of the line is [tex]-1[/tex].
Now, we will substitute [tex]m=-1[/tex] and y-intercept [tex]-9[/tex] in slope form intercept of equation as:
[tex]y=mx+b[/tex], where,
m = Slope,
b = The y-intercept.
[tex]y=-1(x-(-9))[/tex]
[tex]y=-1(x+9)[/tex]
[tex]y=-x-9[/tex]
Therefore, the equation of the line would be [tex]y=-x-9[/tex].
The melons are three for $ 8, how many can you buy for $ 25? Which proportion is correctly stated?
A. 3/8 = 25/x
B. 8/25 = x/3
C. 8/3 = 25/x
D. x/8 = 25/3
Answer:
C
Step-by-step explanation:
The melons are 3 for 8 dollars, this means each melon is worth 8/3 dollars, this is the first proportion in the expression. In this proportion the numerator is the value of money so 25 will be the numerator in the other proportion.
8/3=25/x, C
Find the slope of the line passing through the points (-7,-7) and (-3, 6)
Answer:
13/4
Step-by-step explanation:
The slope of the line between 2 points is found by
m = (y2-y1)/(x2-x1)
= (6--7)/(-3--7)
= (6+7)/(-3+7)
= 13/4
Answer:
The slope is 13/4.
Step-by-step explanation:
Slope formula:
[tex]\displaystyle \frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]\displaystyle \frac{6-(-7)}{(-3)-(-7)}=\frac{13}{4}[/tex]
[tex]\huge\boxed{\frac{13}{4}}[/tex], which is our answer.
Compute the surface integral over the given oriented surface: F=y3i+8j−xk, portion of the plane x+y+z=1 in the octant x,y,z≥0, downward-pointing normal
We are to find the surface integral of [tex]\mathbf{\int \int _S \ F.ds}[/tex]
where;
the surface of the portion of the plane is [tex]x+y+z =1[/tex]; in the 1st octant, [tex]x,y,z \geq 0[/tex] , and:the oriented surface [tex]\mathbf{F = y^3i+8j-xk}[/tex]∴
The plane x + y + z = 1
z = 1 - x - y
[tex]\dfrac{\partial z}{\partial x} = -1[/tex]
[tex]\dfrac{\partial z}{\partial y} = -1[/tex]
Since the surface is oriented downward
∴
[tex]dS = \Big( \dfrac{\partial z}{\partial x}i + \dfrac{\partial z}{\partial y}j - k) dxdy[/tex]
[tex]dS = (-i-j-k) dxdy[/tex]
However,
Flux = [tex]\mathbf{\int \int _S \ F.ds}[/tex]
[tex]= \int \int_R F. \Big( \dfrac{\partial z}{\partial x}i + \dfrac{\partial z}{\partial y}j - k) dxdy \\ \\ \\ = \int^1_{ x=0} \int ^{1-x}_{y=0} ( y^3i + 8j -xk)*(-i-j-k) dx dy \\ \\ \\ =\int^1_{ x=0} \int ^{1-x}_{y=0} (-y^3 -8+x) dxdy \\ \\ \\ = \int^1_{ x=0} \Big( \int ^{1-x}_{y=0} \Big(x-y^3 -8\Big) dy \Big) dx \\ \\ \\ = \int^1_{ x=0} \Bigg [xy - \dfrac{y^4}{4}-8y \Bigg]^{1-x}_{y=0} \ dx \\ \\ \\[/tex]
[tex]= \int^1_{ x=0} \Bigg [x(1-x) - \dfrac{1}{4}(1-x)^{4} - 8(1-x) \Bigg ] dx \\ \\ \\ = \int^1_{ x=0} \Bigg [(x-x^2) - \dfrac{1}{4}(x-1)^4+8(x-1)\Bigg] dx[/tex]
[tex]= \Bigg [ \dfrac{x^2}{2}-\dfrac{x^3}{3} - \dfrac{1}{4} \dfrac{(x-1)^5}{5}+(x-1)^2\Bigg] ^1_0[/tex]
[tex]= \Bigg [ \dfrac{1}{2}-\dfrac{1}{3} -0+0-0+0+ \dfrac{1}{4}\times \dfrac{(-1)^5}{5}-(0-1)^2\Bigg][/tex]
[tex]= \Bigg [ \dfrac{1}{2}-\dfrac{1}{3} - \dfrac{1}{20}-1\Bigg][/tex]
[tex]= \Bigg [ \dfrac{30-20-3-60}{60}\Bigg][/tex]
[tex]= \Bigg [ \dfrac{-53}{60}\Bigg][/tex]
Therefore, we can conclude that the surface integral is [tex]\mathbf{-\dfrac{53}{60}}[/tex]
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To compute a surface integral, take the antiderivatives of both dimensions defining the area with the surface edges as the bounds of the integral. The net flux through the surface can be found using the open surface integral formula.
Explanation:A surface integral over the given oriented surface can be computed by taking the antiderivatives of both dimensions defining the area, with the edges of the surface as the bounds of the integral. The net flux through the surface can be found using the open surface integral formula.
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An item is discounted 20%; the sale price after the discount is $60. What was the original price? Round your answer to the nearest two decimal digits and express your answer without the $ sign (e.g., 1234.25, not $1234.25)
Answer:
75.
Step-by-step explanation:
Let x be the original price. 20% = 0.2. The price with the discount is 60, that is
x-x*0.2 = 60
x(1-0.2) = 60 using common factor,
x (0.8) = 60
x = 60/0.8
x = 75.
So, the original price is $75.
The original price of an item discounted 20% with a sale price after the discount of $60 is $75.
What is discounted price?A discounted price is the marked-down price of an item.
The discounted price represents the selling price after reducing it with the discount.
Data and Calculations:Discount rate = 20%
Discounted price = $60
Original price = $75 ($60/1-20%)
Thus, the original price of an item discounted 20% with a sale price after the discount of $60 is $75.
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Solve the given differential equation by using an appropriate substitution. The DE is of the form dy dx = f(Ax + By + C), which is given in (5) of Section 2.5. dy dx = 4 + y − 4x + 5
No idea what the cited section's method is, but this ODE is linear:
[tex]\dfrac{\mathrm dy}{\mathrm dx}=4+y-4x+5[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}-y=9-4x[/tex]
Multiply both sides by [tex]e^{-x}[/tex] so that the left side can be condensed as the derivative of a product:
[tex]e^{-x}\dfrac{\mathrm dy}{\mathrm dx}-e^{-x}y=(9-4x)e^{-x}[/tex]
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[e^{-x}y\right]=(9-4x)e^{-x}[/tex]
Integrating both sides gives
[tex]e^{-x}y=(4x-5)e^{-x}+C[/tex]
[tex]\implies\boxed{y(x)=4x-5+Ce^x}[/tex]
How much would be in your savings account in eight years after depositing $180 today if the bank pays 8 percent per year? (Do not round intermediate calculations. Round your answer to 2 decimal places.) 10 points Future value Skipped eBook Hint Print Referer LO
Answer:
$333.17
Step-by-step explanation:
Use the compounding formula
[tex]A(t)=P(1+r)^t[/tex]
where A(t) is the amount at the end of the compounding,
P is the initial deposit,
r is the interest rate in decimal form, and
t is the time in years.
Filling in our info:
[tex]A(t)=180(1+.08)^8[/tex]
Simplify a bit to
[tex]A(t)=180(1.08)^8[/tex]
Raise 1.08 to the 8th power and get
A(t) = 180(1.85093021) and then multiply to get
A(t) = $333.17
The monthly utility bills in a city are normally distributed, with a mean of $100 and a standard deviation of $10. If 300 utility bills are randomly selected from this city, approximately how many of them will be more than $115?
First compute the probability that a bill would exceed $115. Let [tex]X[/tex] be the random variable representing the value of a monthly utility bill. Then transforming to the standard normal distribution we have
[tex]Z=\dfrac{X-100}{10}[/tex]
[tex]P(X>115)=P\left(Z>\dfrac{115-100}{10}\right)=P(Z>1.5)\approx0.0668[/tex]
Then out of 300 randomly selected bills, one can expect about 6.68% of them to cost more than $115, or about 20.
The cost (in hundreds of dollars) of tuition at the community college is given by T = 1.25c + 3, where c is the number of credits the student has registered for. If a student is planning to take out a loan to cover the cost of 13 credits, use the model to determine how much money he should borrow.
Answer:
The amount he should borrow is 19.25 hundreds dollars.
Step-by-step explanation:
Given : The cost (in hundreds of dollars) of tuition at the community college is given by [tex]T = 1.25c + 3[/tex], where c is the number of credits the student has registered for. If a student is planning to take out a loan to cover the cost of 13 credits.
To find : Use the model to determine how much money he should borrow?
Solution :
The model is given by [tex]T = 1.25c + 3[/tex]
Where,
c is the number of credits the student has registered.
T is the cost of tuition at the community college.
If a student is planning to take out a loan to cover the cost of 13 credits.
The amount he should borrow will get by putting the value of c in the model,
[tex]T = 1.25c + 3[/tex]
[tex]T = 1.25(13) + 3[/tex]
[tex]T = 16.25 + 3[/tex]
[tex]T =19.25[/tex]
Therefore, The amount he should borrow is 19.25 hundreds dollars.
A supply company manufactures copy machines. The unit cost C (the cost in dollars to make each copy machine) depends on the number of machines made. If x machines are made, then the unit cost is given by the function c(x) = 0.3x^2 - 156x + 26,657 . How many machines must be made to minimize the unit cost?
Do not round your answer.
Answer:
260 machines for minimum cost.
Step-by-step explanation:
c(x) = 0.3x^2 - 156x + 26.657
Finding the derivative:
c'(x) = 0.6x - 156
0.6x - 156 = 0 for maxm/minm cost.
x = 156 / 0.6
= 260
The second derivative is positive (0.6) so this is a minimum.
Answer:
[tex]x=260\ machines[/tex]
Step-by-step explanation:
Note that we have a cudratic function of negative principal coefficient.
The minimum value reached by this function is found in its vertex.
For a quadratic function of the form
[tex]ax ^ 2 + bx + c[/tex]
the x coordinate of the vertex is given by the following expression
[tex]x=-\frac{b}{2a}[/tex]
In this case the function is:
[tex]C(x) = 0.3x^2 - 156x + 26,657[/tex]
So:
[tex]a=0.3\\b=-156\\c=26,657[/tex]
Then the x coordinate of the vertex is:
[tex]x=-\frac{-156}{2(0.3)}[/tex]
[tex]x=260\ machines[/tex]
Then the number of machines that must be made to minimize the cost is:
[tex]x=260\ machines[/tex]
Determine the value of g(4), g(3 / 2), g (2c) and g(c+3) then simplify as much as possible.
g(r) = 2 [tex]\pi[/tex] r h
Answer:
[tex]g(4) = 8 \pi h\\\\g(\frac{3}{2}) = 3 \pi h\\\\ g(2c) = 4 \pi ch\\\\g(c+3) = 2 \pi hc+6\pi h[/tex]
Step-by-step explanation:
You need to substitute [tex]r=4[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:
[tex]g(4) = 2 \pi(4)h\\\\g(4) = 8 \pi h[/tex]
Substitute [tex]r=\frac{3}{2}[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:
[tex]g(\frac{3}{2}) = 2 \pi(\frac{3}{2})h\\\\g(\frac{3}{2}) = 3 \pi h[/tex]
Substitute [tex]r=2c[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:
[tex]g(2c) = 2 \pi(2c))h\\\\g(2c) = 4 \pi ch[/tex]
Substitute [tex]r=c+3[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:
[tex]g(c+3) = 2 \pi (c+3)h\\\\g(c+3) = 2 \pi hc+6\pi h[/tex]
For this case we have the following function:
[tex]g (r) = 2 \pi * r * h[/tex]
We must evaluate the function for different values:
[tex]g (4) = 2 \pi * (4) * h = 8 \pi*h\\g (\frac {3} {2}) = 2 \pi * (\frac {3} {2}) * h = 3 \pi*h\\g (2c) = 2 \pi * (2c) * h = 4 \pi * c * h\\g (c + 3) = 2 \pi * (c + 3) * h = 2 \pi * c * h + 6 \pi * h[/tex]
Answer:
[tex]g (4) = 8 \pi*h\\g (\frac {3} {2}) =3 \pi*h\\g (2c) = 4 \pi * c * h\\g (c + 3) = 2 \pi * c * h + 6 \pi * h[/tex]
Suppose that you follow a population over time. When you plot your data on a semilog plot (using logs with base 10), a straight line with slope 0.1 results. Furthermore, assume that the population size at time 0 was 80. What function best describes the population size at time t?
Answer:
[tex]P(t)=80(10^{0.1t})[/tex]
Step-by-step explanation:
The 'y' axis represent log(P), so it may be modeled as a line (or linear function), where its slope is 0.1:
[tex]log(P)=0.1t+C[/tex]
Pow each part of the equation by 10:
[tex]10^{log(P)}=10^{0.1t+C}\\ P=10^{0.1t+C}[/tex]
Evaluate at t=0, where the population is known.
[tex]P(0)=10^{C}=80[/tex]
Applying logarithmic properties:
[tex]P=10^{0.1t+C}=10^{0.1t}*10^{C}[/tex]
So, the final function is:
[tex]P(t)=80(10^{0.1t})[/tex]
The population size over time follows an exponential growth function described by P(t) = 80 * [tex]e^(0.23026t)[/tex], where the slope on a semilog plot of 0.1 is converted to the natural log base to find the growth rate constant.
When dealing with population growth, plotting the data on a semilog plot where the logarithm of the population size is plotted against time can simplify the analysis. If the resulting plot is a straight line with a slope of 0.1, this indicates exponential growth, because plotting an exponential function on a semilog plot yields a straight line.
The general form of an exponential growth function is:
P(t) = P0 * [tex]e^{kt}[/tex]Where:
P(t) is the population at time tP0 is the initial population sizek is the growth rate constantt is timeGiven that the slope of the line on the semi log plot is 0.1, this slope represents the constant k in the context of logs with base 10.
To convert this to a natural logarithm base (e), use the conversion factor ln(10):
k = 0.1 * ln(10)
Since ln(10) ≈ 2.3026, we have:
k = 0.1 * 2.3026
≈ 0.23026
Given the initial population size (P0) at time t = 0 is 80, the function describing the population size over time t is:
P(t) = 80 * [tex]e^(0.23026t)[/tex],
Solve the congruence 2x 3 (mod 11):
[tex]2x\equiv3\pmod{11}[/tex]
Since 3 * 4 = 12 = 1 mod 11, we can multiply both sides by 4 to get
[tex]4\cdot2x\equiv3\cdot4\pmod{11}\implies8x\equiv1\pmod{11}[/tex]
Since 8 * 7 = 56 = 1 mod 11, multiplying both sides by 7 gives
[tex]7\cdot8x\equiv7\cdot1\pmod{11}\implies x\equiv7\pmod{11}[/tex]
A group of research proposals was evaluated by a panel of experts to decide whether or not they were worthy of funding. When these same proposals were submitted to a second independent panel of experts, the decision to fund was reversed in 30% of the cases. If the probability that a proposal is judged worthy of funding by the first panel is .2, what are the probabilities that: a.A worthy proposal is approved by both panels. b.A worthy proposal is disapproved by both panels. c.A worthy proposal is approved by one panel.
Explanation of probabilities for a worthy research proposal approved by two panels, disapproved by two panels, and approved by one panel.
A worthy proposal is approved by both panels:
Probability = (Probability approved by Panel 1) × (Probability approved by Panel 2) = 0.2 × 0.7 = 0.14
A worthy proposal is disapproved by both panels:
Probability = (Probability disapproved by Panel 1) × (Probability disapproved by Panel 2) = 0.8 × 0.3 = 0.24
A worthy proposal is approved by one panel:
Probability = (Probability approved by Panel 1 and disapproved by Panel 2) + (Probability disapproved by Panel 1 and approved by Panel 2) = (0.2 × 0.3) + (0.8 × 0.7) = 0.06 + 0.56 = 0.62
Use the arc length formula to find the length of the curve y = 4x − 5, −1 ≤ x ≤ 3. Check your answer by noting that the curve is a line segment and calculating its length by the distance formula.
Answer:
[tex]4\sqrt{17}[/tex]
Step-by-step explanation:
Let's find the answer by using the arc length formula which is:
[tex]\int\limits^a_b {\sqrt{1+(\frac{dy}{dx})^{2} } } \, dx[/tex]
First, let's find dy/dx which is:
y=4x-5
y'=4*(1)-0
y'=4, now let's use the formula:
[tex]\int\limits^3_{-1} {\sqrt{1+4^{2}} } \, dx=\sqrt{17} *(3-(-1))=4\sqrt{17}[/tex]
Now, using the distance formula we have:
[tex]d=\sqrt{(x2-x1)^{2} +(y2-y1)^{2} }[/tex]
[tex]y(-1)=4*(-1)-5=-9 \\y(3)=4*(3)-5=7[/tex]
So we have two points (-1, -9) and (3, 7) so:
[tex]d=\sqrt{(3-(-1))^{2} +(7-(-9))^{2} }=4\sqrt{17}[/tex]
Notice both equations gave the same length [tex]4\sqrt{17}[/tex].
Final answer:
To find the length of the curve y = 4x - 5, -1 ≤ x ≤ 3 using the arc length formula, integrate sqrt(1 + (dy/dx)^2) from x = -1 to x = 3. The length of the curve is 4√17. The result can be confirmed by calculating the length using the distance formula.
Explanation:
To find the length of the curve y = 4x - 5, -1 ≤ x ≤ 3 using the arc length formula, we need to integrate the square root of 1 + (dy/dx)^2 from x = -1 to x = 3. The derivative of y = 4x - 5 is dy/dx = 4. Substituting this into the arc length formula, we have:
L = ∫sqrt(1 + (dy/dx)^2) dx = ∫sqrt(1 + 4^2) dx = ∫sqrt(17) dx = x√17 + C
Now, plugging in the limits of integration, we have:
L = [(3√17) + C] - [(-1√17) + C] = (3√17) - (-1√17) = 4√17
To check our answer, we can use the distance formula. The endpoints of the line segment are (-1, -9) and (3, 7). Using the distance formula:
D = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((3 - (-1))^2 + (7 - (-9))^2) = sqrt(4^2 + 16^2) = sqrt(272) = 16√17
As we can see, the length of the curve obtained using the arc length formula (4√17) matches the length calculated using the distance formula (16√17), confirming our answer.
A nurse must infuse 1.5 ml of solution in x minutes and she has 650 ml of solution how many minutes will it take for the medicine to be given
Answer:
433 minutes
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.
The nurse is infusing 1.5 ml of solution every x minutes, Since there are a total of 650 ml we can divide the amount the nurse is infusing every x minutes by the total in order to calculate the amount of time it will take the nurse to finish.
[tex]\frac{650}{1.5} = 433.33min[/tex]
So we can see that it will take the nurse 433 min to infuse all of the solution.
Answer:
In 433.33 minutes will it take for the medicine to be given.
Step-by-step explanation:
Given : A nurse must infuse 1.5 ml of solution in x minutes and she has 650 ml of solution.
To find : How many minutes will it take for the medicine to be given ?
Solution :
As 1.5 ml of solution infuse in x minutes
i.e. the amount of solution she has = [tex]1.5x[/tex]
Now, she has 650 ml of solution.
which means [tex]1.5x=650[/tex]
Solving the equation,
[tex]x=\frac{650}{1.5}[/tex]
[tex]x=433.33[/tex]
Therefore, in 433.33 minutes will it take for the medicine to be given.
Assume that women's heights are normally distributed with a mean given by mu equals 62.5 in, and a standard deviation given by sigma equals 2.5 in. (a) If 1 woman is randomly selected, find the probability that her height is less than 63 in. (b) If 35 women are randomly selected, find the probability that they have a mean height less than 63 in. (a) The probability is approximately nothing. (Round to four decimal places as needed.) (b) The probability is approximately nothing. (Round to four decimal places as needed.)
Answer: a) The probability is approximately = 0.5793
b) The probability is approximately=0.8810
Step-by-step explanation:
Given : Mean : [tex]\mu= 62.5\text{ in}[/tex]
Standard deviation : [tex]\sigma = \text{2.5 in}[/tex]
a) The formula for z -score :
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
Sample size = 1
For x= 63 in. ,
[tex]z=\dfrac{63-62.5}{\dfrac{2.5}{\sqrt{1}}}=0.2[/tex]
The p-value = [tex]P(z<0.2)=[/tex]
[tex]0.5792597\approx0.5793[/tex]
Thus, the probability is approximately = 0.5793
b) Sample size = 35
For x= 63 ,
[tex]z=\dfrac{63-62.5}{\dfrac{2.5}{\sqrt{35}}}\approx1.18[/tex]
The p-value = [tex]P(z<1.18)[/tex]
[tex]= 0.8809999\approx0.8810[/tex]
Thus , the probability is approximately=0.8810.