Rolle's theorem works for a function [tex]f(x)[/tex] over an interval [tex][a,b][/tex] if:
[tex]f(x)[/tex] is continuous on [tex][a,b][/tex][tex]f(x)[/tex] is differentiable on [tex](a,b)[/tex][tex]f(a)=f(b)[/tex]This is our case: [tex]f(x)[/tex] is a polynomial, so it is continuous and differentiable everywhere, and thus in particular it is continuous and differentiable over [0,4].
Also, we have
[tex]f(0)=7=f(4)[/tex]
So, we're guaranteed that there exists at least one point [tex]c\in(a,b)[/tex] such that [tex]f'(c)=0[/tex].
Let's compute the derivative:
[tex]f'(x)=3x^2-2x-12[/tex]
And we have
[tex]f'(x)=0 \iff x= \dfrac{1\pm\sqrt{37}}{3}[/tex]
In particular, we have
[tex]\dfrac{1+\sqrt{37}}{3}\approx 2.36[/tex]
so this is the point that satisfies Rolle's theorem.
The number C that satisfies the conclusion of Rolle's Theorem on the interval [0, 4] is: [tex]\[ c = \frac{1 + \sqrt{37}}{3} \][/tex]
To verify that the function [tex]\( f(x) = x^3 - x^2 - 12x + 7 \)[/tex] satisfies the three hypotheses of Rolle's Theorem on the interval [0, 4] and then to find all numbers c that satisfy the conclusion of Rolle's Theorem, follow these steps:
1. The function f is continuous on the closed interval [a, b]:
- [tex]\( f(x) = x^3 - x^2 - 12x + 7 \)[/tex] is a polynomial, and polynomials are continuous everywhere.
- Therefore, f is continuous on [0, 4].
2. The function f is differentiable on the open interval (a, b):
- Again, [tex]\( f(x) = x^3 - x^2 - 12x + 7 \)[/tex] is a polynomial, and polynomials are differentiable everywhere.
- Therefore, f is differentiable on (0, 4).
3. f(a) = f(b) :
- Calculate [tex]\( f(0) \)[/tex] and f(4):
[tex]\[ f(0) = 0^3 - 0^2 - 12 \cdot 0 + 7 = 7 \] \[ f(4) = 4^3 - 4^2 - 12 \cdot 4 + 7 = 64 - 16 - 48 + 7 = 7 \][/tex]
- Therefore, f(0) = f(4) = 7 .
Since all three hypotheses are satisfied, by Rolle's Theorem, there exists at least one number c in (0, 4) such that f'(c) = 0 .
Finding c
1. Compute the derivative of f:
[tex]\[ f(x) = x^3 - x^2 - 12x + 7 \] \[ f'(x) = 3x^2 - 2x - 12 \][/tex]
2. Set the derivative equal to zero and solve for x:
[tex]\[ f'(x) = 3x^2 - 2x - 12 = 0 \][/tex]
Solve the quadratic equation using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where a = 3, b = -2 , and c = -12 :
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-12)}}{2 \cdot 3} \] \[ x = \frac{2 \pm \sqrt{4 + 144}}{6} \] \[ x = \frac{2 \pm \sqrt{148}}{6} \] \[ x = \frac{2 \pm 2\sqrt{37}}{6} \] \[ x = \frac{1 \pm \sqrt{37}}{3} \][/tex]
3. Check which solutions are in the interval (0, 4):
- For [tex]\( x = \frac{1 + \sqrt{37}}{3} \)[/tex]:
[tex]\[ \frac{1 + \sqrt{37}}{3} \approx \frac{1 + 6.08}{3} \approx \frac{7.08}{3} \approx 2.36 \][/tex]
- For [tex]\( x = \frac{1 - \sqrt{37}}{3} \)[/tex]:
[tex]\[ \frac{1 - \sqrt{37}}{3} \approx \frac{1 - 6.08}{3} \approx \frac{-5.08}{3} \approx -1.69 \][/tex]
- This solution is not in the interval (0, 4).
A 3-card hand is dealt from a standard 52-card deck, and then one of the 3 cards is chosen at random. If only one of the cards in the hand is a 3, what is the probability that the chosen card is a 3?
The probability that the chosen card is a 3 is?
(Simplify your answer. Type an integer or a fraction.)
Answer:
1/3
Step-by-step explanation:
Once you know there is a 3 in the hand, and only one card is drawn from the cards in the hand, then the probability of drawing the 3 is 1/3.
Answer:
P (chosen card is 3) = [tex]\frac{1}{3}[/tex]
Step-by-step explanation:
We are given that a 3-card hand is dealt from a standard 52-card deck, and then one of the 3 cards is chosen at random.
Given that only one of the cards is a 3, we are to find the probability that the chosen card is a 3.
Since only one card is a 3 and one card is chosen so:
P (chosen card is 3) = 1 / 3
According to one pollster, 43 % of people are afraid of flying. Suppose that a sample of size 26 is drawn. Find the value of standard error , the standard deviation of the distribution of sample proportions.
Answer: 0.0971
Step-by-step explanation:
Given : Sample size : [tex]n=26[/tex]
The percent of people are afraid of flying [tex]=43\%[/tex]
Thus the proportion of people are afraid of flying [tex]P=0.43[/tex]
We know that the formula to find the standard deviation of the distribution of sample proportions is given by :-
[tex]\text{S.E.}=\sqrt{\dfrac{P(1-P)}{n}}\\\\\Rightarrow\text{S.E.}=\sqrt{\dfrac{0.43(1-0.43)}{26}}\\\\\Rightarrow\ \text{S.E.}=0.0970923430396\approx0.0971[/tex]
Hence, the standard deviation of the distribution of sample proportions = 0.0971
The standard error for the distribution of sample proportions is approximately 0.0966 (rounded to four decimal places).
The standard error (SE) for a sample proportion can be calculated using the following formula:
SE = √[p(1 - p) / n]
Where:
p is the population proportion (0.43, or 43% expressed as a decimal).
n is the sample size (26).
Let's calculate the standard error:
SE = √[0.43 * (1 - 0.43) / 26]
SE = √[0.43 * 0.57 / 26]
SE = √(0.009351923076923077)
SE ≈ 0.096645
So, the standard error for the distribution of sample proportions is approximately 0.0966 (rounded to four decimal places).
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Show that the following number is rational: 7.526262626... show all work, discrete math
[tex]x=7.5\overline{26}\\10x=75.\overline{26}\\1000x=7526.\overline{26}\\1000x-10x=7526.\overline{26}-75.\overline{26}\\990x=7451\\x=\dfrac{7451}{990}[/tex]
It can be expressed as a fraction with integer numerator and denominator, so it's a rational number.
A batch of one hundred cars is inspected by testing four randomly selected cars. If one of the four is defective, the batch is rejected. What is the probability that the batch is accepted if it contains five defective cars?
Answer: The probability that the batch is accepted is 0.812.
Step-by-step explanation:
Since we have given that
Number of defective cars = 5
Total number of cars = 100
Probability of getting first car non defective = [tex]\dfrac{95}{100}[/tex]
probability of getting second car non defective = [tex]\dfrac{94}{99}[/tex]
Probability of getting third car non defective = [tex]\dfrac{93}{98}[/tex]
Probability of getting fourth car non defective = [tex]\dfrac{92}{97}[/tex]
Using the multiplication rule, probability that the batch is accepted is given by
[tex]\dfrac{95}{100}\times \dfrac{94}{99}\times \dfrac{93}{98}\times \dfrac{92}{97}\\\\=0.812[/tex]
Hence, the probability that the batch is accepted is 0.812.
In order to estimate the mean amount of time computer users spend on the internet each month, how many computer users must be surveyed in order to be 95% confident that your sample mean is within 12 minutes of the population mean? Assume that the standard deviation of the population of monthly time spent on the internet is 213 min. What is a major obstacle to getting a good estimate of the population mean? Use technology to find the estimated minimum required sample size. The minimum sample size required is 1211 computer users. (Round up to the nearest whole number.) What is a major obstacle to getting a good estimate of the population mean?
PLEASE HELP!!
Demonstrate your understanding of how to solve exponential equations by rewriting the base. Solve the problem below fully and explain all the steps...
25^3k = 625
To solve the equation [tex]25^3^k[/tex] = 625, rewrite the base as a power of 5 and use the exponentiation rule. Simplify the equation and equate the exponents to solve for k.
Explanation:To solve the exponential equation 253k = 625, we need to rewrite the base of 25 as a power of 5. Since 25 = 52, we can rewrite the equation as (52)3k = 625.
Using the rule (ab)c = ab × c, we can simplify the equation to 52 × 3k = 625.
Now, we can rewrite 625 as a power of 5 by realizing that 625 = 54. Therefore, we have 52 × 3k = 54.
Since the bases are the same, we can equate the exponents and solve for k:
2 × 3k = 4
6k = 4
k = 4/6
k = 2/3
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To solve the exponential equation 25^3k = 625, rewrite the base 25 as a power of 5. Simplify the equation and set the exponents equal to each other to solve for k.
Explanation:To solve the exponential equation 253k = 625, we can rewrite the base 25 as a power of 5, since 52 = 25. So, the equation becomes (52)3k = 625. Using the rule of exponents (am)n = amn , we can simplify it to 56k = 625.
Next, we can rewrite 625 as a power of 5: 625 = 54. So, the equation becomes 56k = 54.
Since the bases are the same, we can set the exponents equal to each other: 6k = 4. Solving for k, we divide both sides by 6 to get k = 4/6 = 2/3.
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An urn contains 4 white and 4 black balls. We randomly choose 4 balls. If 2 of them are white and 2 are black, we stop. If not, we replace the balls in the urn and again randomly select 4 balls. This continues until exactly 2 of the 4 chosen are white. What is the probability that we shall make exactly n selections?
Answer:
The probability that we shall make exactly n selections is [tex]P(X = n)=(\frac{17}{35})^{n-1}\frac{18}{35}[/tex].
Step-by-step explanation:
It is given that an urn contains 4 white and 4 black balls and we randomly choose 4 balls. If 2 of them are white and 2 are black, we stop.
The total number of ways to select exactly 2 white and 2 black balls.
[tex]^4C_2\times ^4C_2=\frac{4!}{2!(4-2)!}\times \frac{4!}{2!(4-2)!}=6\times 6=36[/tex]
The total number of ways to select 4 balls from 8 balls is
[tex]^8C_4=\frac{8!}{4!(8-4)!}=\frac{8\times 7\times 6\times 5\times 4!}{4\times 3\times 2\times 1\times !4!}=70[/tex]
The probability of selecting exactly 2 white and 2 black balls is
[tex]p=\frac{36}{70}=\frac{18}{35}[/tex]
The probability of not selecting exactly 2 white and 2 black balls is
[tex]q=1-p=1-\frac{18}{35}=\frac{17}{35}[/tex]
If we not get exactly 2 white and 2 black balls, then we replace the balls in the urn and again randomly select 4 balls.
The probability that we shall make exactly n selections is
[tex]P(X = n)=(q)^{n-1}p[/tex]
[tex]P(X = n)=(\frac{17}{35})^{n-1}\frac{18}{35}[/tex]
Therefore the probability that we shall make exactly n selections is [tex]P(X = n)=(\frac{17}{35})^{n-1}\frac{18}{35}[/tex].
_________________is a technology that combines data from one or more sources so it can be compared for making business decisions
A. Data architect
B. Data warehouse
C. Data management
D. Data architecture
Answer:
Data warehouse
Step-by-step explanation:
Data warehouse is a technology that combines data from one or more sources so it can be compared for making business decisions.
Moreover, Data warehouse can be explained as a technology that combines from one or more sources so it can be prepared for making business decisions. A data warehouse is constructed by Integrating data from multiple heterogeneous sources that support analytical reporting, and decision making.
Heather has $45.71 in her savings account. She bought six packs of markers to donate to her school. Write an expression for how much money she has in her bank account after the donation
HAS TO MATCH ONE OF THOSE
A. 45.71+6m
B. 45.71−6
C. 45.71+6
D. 45.71−6m
Answer:
D. 45.71 - 6m
Step-by-step explanation:
Let m = the cost of a marker
Then 6m = the cost of six markers
Heather is paying for these from her savings account.
After deducting the cost of the markers, the amount in her account will be
45.71 - 6m
Solve the congruence 9x 17 (mod 26).
We have [tex]17\equiv-9\pmod{26}[/tex], so that [tex]x\equiv-1\pmod{26}[/tex], so [tex]x\equiv25\pmod{26}[/tex], and any solution of the form [tex]x=25+26n[/tex] satisfies the congruence, where [tex]n[/tex] is any integer.
The question involves solving a congruence equation. Given the equation 9x + 17 ≡ 0 (mod 26), we would typically isolate x to find the solution. However, the question seems to be missing an operator. Hence, an explicit answer cannot be provided.
Explanation:The question involves solving a congruence. To solve the accord 9x ≡ 17 (mod 26), you must find an integer x such that 9x leaves a remainder of 17 when divided by 26.
This unity can be rewritten as 9x - 17 = 26k, where k is an integer. Our task is to solve for x given these parameters.
Given the nature of the question, I cannot provide a direct solution because it is missing an operator between 9x and 17. Assuming the operator is '+', the congruence will be 9x + 17 ≡ 0 (mod 26).
The steps to solve a congruence equation can vary, but generally, the goal is to isolate x on one side of the equation. However, it's easier to proceed with this congruence with explicit details.
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Write an equation for the line passing through the given point and having the given slope.
(10,6), m= 1
The equation of the line is:
(Type your answer in slope-iritercept form. Use integers or fractions for any numbers in the equation.)
Answer: The equation, in "slope-intercept form" ; is:
________________________________________________
→ " y = x + 4 " .
________________________________________________
Step-by-step explanation:
________________________________________________
Use the formula for linear equations; written in "point-slope format" ;
which is:
y − y₁ = m( x − x₁ ) ;
We are given the slope, "m" ; has a value of: "1 " ;
that is; " m = 2 " .
________________________________________________
We are given the coordinates to 1 (one) point on the line; in which the coordinates are in the form of :
" ( x₁ , y₁ ) " ;
→ that given point is: "(10, 6)" ;
in which: x₁ = 10 ;
y₁ = 6 .
→ Given: The slope, "m" equals "1" ; ________________________________________________
Let's plug our known values into the formula:
→ " y − y₁ = m( x − x₁ ) " ;
_______________________________________________
→ As follows:
→ " y − 10 = 1(x − 6) ;
______________________________________________
Now, focus on the "right-hand side of the equation" ;
→ 1(x − 6) = ? ; Simplify.
______________________________________________
Note the "distributive property" of multiplication:
→ a(b + c) = ab + ac ;
As such: " 1(x − 6) = (1*x) + (1 * -6) " ;
= 1x + (-6) ;
= x − 6 ;
[Note that: " 1 x = 1 * x = x " ;
[Note that " + (-6) " = " ( " − 6 " ) .] ;
→ {since: "Adding a negative" is the same as:
"subtracting a positive."} ;
________________________________________________
Now, let us bring down the "left-hand side of the equation" ; &
rewrite the entire equation; as follows:
________________________________________________
→ " y − 10 = x − 6 " ;
________________________________________________
Note: We want to rewrite the equation in "slope-intercept form" ;
that is; " y = mx + b " ;
in which: "y" ; stands alone as a single variable on the "left-hand side" of the equation; with "no coefficients" [except for the "implied coefficient" of " 1 "} ;
"m" is the coefficient of "x" ;
and the "slope of the line" ;
Note that "m" may be a "fraction or decimal" ; and may be "positive or negative.
If the slope is "1" ; (that is "1 over 1" ; or: "[tex]\frac1}{1}[/tex]" ;
then, " m = 1 " ; and we can write " 1x " as simply "x" ; since the implied coefficient is "1" ;
→ since " 1" , multiplied by any value {in our case, any value for "x"} , equals that same value.
________________________________________________
"b" refers to the "y-intercept" of the graph of the equation;
that is; the "y-value" of the point at which the graphed line of the equation crosses the "y-axis" ;
that is, the "y-value" of the coordinates of the point at which the graphed line of the equation crosses the "y-axis" ;
that is, the ["y-value" of the] y-intercept" .
Note that the value of "b" may be positive or negative, and may be a decimal or fraction.
If the value for "b" is negative, the equation can be written in the form:
" y = mx - b " ;
{since: " y = mx + (-b) " is a bit tedious .}
If the y-intercept is "0" ; (i.e. the line crosses the y-axis at the origin, at point: " (0,0) " ;
then we simply write the equation as: "y = mx " ;
________________________________________________
So; we have: → " y − 10 = x − 6 " ;
________________________________________________
→ We want to rewrite our equation in slope-intercept form,
that is; " y = mx + b " ; as explained above.
We can add "10" to each side of the equation ; to isolation the "y" on the "left-hand side" of the equation:
→ " y − 10 + 10 = x − 6 + 10 " ;
to get:
→ " y = x + 4 " ;
________________________________________________
→ which is our answer.
________________________________________________
Note: This answer: " y = x + 4 " ;
→ is written in the "slope-intercept format";
→ " y = mx + b " ;
in which: "y" is isolated as a single variable on the "left-hand side of the equation" ;
The slope of the equation is "1" ; or an implied value of "1" ;
that is; " m = 1 " ;
"b = 4 " ;
→ {that is; the "y-value" of the "y-intercept" — "(0, 4)" — of the graph of the equation is: "4 ".} .
________________________________________________
Hope this answer is helpful!
Best wishes to you in your academic pursuits
— and within the "Brainly" community!
________________________________________________
pH measurements of a chemical solutions have mean 6.8 with standard deviation 0.02. Assuming all pH measurements of this solution have a nearly symmetric/bell-curve distribution. Find the percent (%) of pH measurements reading below 6.74 OR above 6.76.
Answer: 2.14 %
Step-by-step explanation:
Given : pH measurements of a chemical solutions have
Mean : [tex]\mu=6.8[/tex]
Standard deviation : [tex]\sigma=0.02[/tex]
Let X be the pH reading of a randomly selected customer chemical solution.
We assume pH measurements of this solution have a nearly symmetric/bell-curve distribution (i.e. normal distribution).
The z-score for the normal distribution is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x = 6.74
[tex]z=\dfrac{6.74-6.8}{0.02}=-3[/tex]
For x = 6.76
[tex]z=\dfrac{6.76-6.8}{0.02}=-2[/tex]
The p-value =[tex]P(6.74<x<6,76)=P(-3<z<-2)[/tex]
[tex]P(z<-2)-P(z<-3)=0.0227501- 0.0013499=0.0214002\approx0.0214[/tex]
In percent, [tex]0.0214\times=2.14\%[/tex]
Hence, the percent of pH measurements reading below 6.74 OR above 6.76 = 2.14%
Student grades on a chemistry exam were:77, 79, 76, 82, 86, 50, 79, 81, 83, 99
Construct a stem-and-leaf plot of the data. (Use the tens digit as the stem and the ones digit as the leaf. Enter your answers from smallest to largest, separated by spaces. Enter NONE for stems with no leaves.)
A stem-and-leaf plot of the chemistry exam scores is created by dividing each score into a tens digit (the stem) and a ones digit (the leaf). We then group the data by stem and list the leaves for each stem, giving us a distribution of the exam scores.
Explanation:To construct a stem-and-leaf plot, you can follow this process:
Organize the data from least to greatest. Doing so, we get: 50, 76, 77, 79, 79, 81, 82, 83, 86, 99.Divide each number into a stem and a leaf, where the stem is the tens place and the leaf is the ones digit.So, our stem-and-leaf plot would look like this:
5 | 0
7 | 6 7 9 9
8 | 1 2 3 6
9 | 9
In this plot, for example, '7 | 6 7 9 9' means there are scores of 76, 77, 79, and 79.
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What is the area of the composite figure?
A (8π + 6) in.2
B (8π + 12) in.2
C (8π + 18) in.2
D (8π + 24) in.2
Answer:
B. (8π + 12) in²
Step-by-step explanation:
1. Identify the formula for the area of both a triangle and a circle.
Triangle: 1/2(b)(h)
b = base
h = height
Circle: πr²
r = radius
2. Start by finding the area of the circle, since we already have all the needed information for the variables in the equation.
π(4)² → π(16) → 16π
3. Half the answer we just got as the area of the circle. We are doing this because we only have half a circle in the diagram, and we solved for the area of a full circle.
(16π)/2 → 8π
4. Next find the base of the triangle, since this is the only information we do not yet have for the triangle. We will find this by doubling the 4, since 4 inches is only half the length of the base.
4 × 2 = 8
5. Plug all the information of the triangle into the area of a triangle formula and solve.
1/2(8)(3) → 1/2(24) → 12
6. Add both the area of the semi-circle and triangle together because they are one shape that we are finding the area for.
8π + 12
7. Label answer with units of measurement
(8π + 12) in²
ANSWER
The correct option is B
EXPLANATION
The composite figure is made up of a semicircle and an isosceles triangle.
The area of a semicircle is
[tex] \frac{1}{2}\pi {r}^{2} [/tex]
From the diagram, the radius is
[tex]r = 4 \: in[/tex]
When we substitute, area of the semicircle is
[tex] \frac{1}{2} \times \pi \times {4}^{2} [/tex]
[tex]\frac{1}{2} \times \pi \times 16[/tex]
[tex]8\pi \: \: {in}^{2} [/tex]
The area of the isosceles triangle is
[tex] \frac{1}{2} \times base \times height[/tex]
[tex] = \frac{1}{2} \times (4 + 4) \times 3[/tex]
[tex] = \frac{1}{2} \times 8 \times 3[/tex]
[tex] = 12 \: {in}^{2} [/tex]
We add the two areas to obtain the area of the composite figure to be:
[tex](8\pi + 12) {in}^{2} [/tex]
how many $50 bills is found in $890
Answer:
17 bills
Step-by-step explanation:
There are 17 $50 bills is found in $890.
All you have to do is:
890 ÷ 50
However, that would equal 17.8, which is not a whole number. Therefore, there are only 17 $50 bills found in $890.
For a binomial probability distribution, it is unusual for the number of successes to be less than μ − 2.5σ or greater than μ + 2.5σ. (a) For a binomial experiment with 10 trials for which the probability of success on a single trial is 0.2, is it unusual to have more than five successes? Explain
Answer:
no of success deemed to be usual 5
that is more than 5 success unusual
Step-by-step explanation:
Given data
number of successes = less than μ − 2.5σ
number of successes = greater than μ + 2.5σ
trials n = 10
probability single trial = 0.2
to find out
is it unusual to have more than five successes
solution
we can say that
mean of the binomial, distribution that is
mean = probability single trial × trials n
mean = 10 × .2
mean = 2
and standard deviation = √(mean× (1-probability))
standard deviation = √2× (1-0.2))
standard deviation = 1.2649
so no of successes are
= μ − 2.5σ and = μ + 2.5σ
= 2 − 2.5(1.2649) and = 2 + 2.5(1.2649)
= -0.16225 and = 5.16225
so now we say no of success deemed to be usual 5
that is more than 5 success unusual here
To determine whether it is unusual to have more than five successes in a binomial experiment with 10 trials and a probability of success of 0.2, we can use the mean and standard deviation of the binomial distribution. The probability of having six or more successes in this experiment is quite low (less than 0.05), so it would be considered unusual.
Explanation:To determine whether it is unusual to have more than five successes in a binomial experiment with 10 trials and a probability of success of 0.2, we can use the mean and standard deviation of the binomial distribution. The mean is found by multiplying the number of trials by the probability of success (10 * 0.2 = 2), and the standard deviation is found using the formula √(npq) (where q = 1 - p). For this experiment, the standard deviation would be √(10 * 0.2 * 0.8) = 1.26.
To determine whether it is unusual to have more than five successes, we need to find the probability of having six or more successes. We can use the cumulative binomial distribution to find this probability. Using a calculator or a statistical software, we can find that the probability is approximately 0.003.
Since the probability is quite low (less than 0.05), we can consider it unusual to have more than five successes in this particular binomial experiment.
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1 inch = 2.54 centimeters
800 centimeters= _______ inches
please round to nearest tenth please
I have been getting wrong answers
Answer:
It should be 314.9
Step-by-step explanation:
In every centimeter, there are about .3937 inches
So if you multiple .3937 by 800 and round to the nearest tenth, you get that answer
Allison is 26 years old and plans to retire at age 65 with $1,90,000 in her retirement account. What amount would she have to set aside now in an investment paying 7% annual interest if the compounding is done daily (assume 365 days in a year)?
Final answer:
Allison would need to set aside approximately $21,338.60 in the investment now.
Explanation:
To calculate the amount Allison would have to set aside now in an investment paying 7% annual interest with daily compounding, we can use the formula for compound interest:
[tex]A = P(1 + r/n)^(nt)[/tex]
Where:
A = the final amount, which is $1,90,000 in this caseP = the principal amount (the amount to be set aside now)r = the annual interest rate, which is 7%n = the number of times interest is compounded per year, which is 365 in this caset = the number of years, which is 65 - 26 = 39Plugging in the values, we get:
[tex]$1,90,000 = P(1 + 0.07/365)^(365 * 39)[/tex]
Simplifying the equation:
[tex]P = $1,90,000 / (1 + 0.07/365)^(365 * 39)[/tex]
Calculating this using a scientific calculator or software, Allison would need to set aside approximately $21,338.60 in the investment now.
Solve the problem. Determine which of the following sets is a subspace of Pn for an appropriate value of n. A: All polynomials of the form p(t) = a + bt2, where a and b are in ℛ B: All polynomials of degree exactly 4, with real coefficients C: All polynomials of degree at most 4, with positive coefficients
None of the given sets A, B, or C is a subspace of Pn. Set A does not always include the zero vector, set B does not contain the zero polynomial, and set C excludes cases where coefficients aren't positive.
Explanation:In mathematics, a subspace must satisfy three conditions: it must contain the zero vector, it must be closed under addition, and it must be closed under scalar multiplication. Looking at the set A, we see that it does not contain all polynomials of the form p(t) = a + bt2, as it won't include the zero vector when both a and b are not zero. Therefore, set A is not a subspace.
Set B includes all polynomials of exactly degree 4. But it doesn't contain the zero polynomial which is of degree 0, so B isn't a subspace either.
As for set C, although it includes polynomials of degree at most 4, it requires the coefficients to be positive which is problematic in case of scalar multiplication. Hence, C is not a subspace.
Therefore, none of the specified sets is a subspace of Pn for the appropriate value of n.
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The parametric equations x = x1 + (x2 − x1)t, y = y1 + (y2 − y1)t where 0 ≤ t ≤ 1 describe the line segment that joins the points P1(x1, y1) and P2(x2, y2). Draw the triangle with vertices A(1, 1), B(4, 3), C(1, 7). Find the parametrization, including endpoints, and sketch to check. (Enter your answers as a comma-separated list of equations. Let x and y be in terms of t.)
Answer:
see below
Step-by-step explanation:
Filling the given numbers into the given formulas, you have ...
Line AB:
x = 1 +(4-1)t, y = 1 +(3-1)t
x = 1+3t, y = 1+2t . . . . . . simplify
Line BC:
x = 4 +(1-4)t, y = 3 +(7-3)t
x = 4 -3t, y = 3 +4t . . . . . simplify
Line AC:
x = 1 +(1-1)t, y = 1 +(7-1)t
x = 1, y = 1+6t . . . . . . . . . .simplify
A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container. Be sure to justify that the cost you find is the cheapest cost for this container
Answer:163.54$
Step-by-step explanation:
Given data
Volume of Storage(V)=[tex]{10m^3}[/tex]
Length=2breadth
Let Length be L,Breadth be & height be H
therefore
10=LBH
Now substitutes the values
10=2[tex]{B^2}H[/tex]
5=[tex]{B^2}H[/tex]
Now cost for base is [tex]{C_1}=2{B^2}\times10[/tex]
Cost for side walls is[tex]{C_2}={2LH}\times6+2BH}\times6[/tex]
Now total cost(C)=[tex]{C_1}+{C_2}[/tex]
C=20[tex]{B^2}H[/tex]+[tex]{2LH}\times6[/tex]+[tex]2BH}\times6[/tex]
C=20[tex]{B^2}H[/tex]+24BH+[tex]12BH[/tex]
C=[tex]20{B^2}+36B\times\frac{5}{B^{2}}[/tex]
Now Differentiating With respect to Breadth to get minimum cost
[tex]\frac{\mathrm{d} C}{\mathrm{d} B}=0[/tex]
[tex]we\ get\ B=\sqrt[3]{4.5}=1.65m[/tex]
[tex]L=3.30m[/tex]
[tex]H=1.836m[/tex]
and mimimum cost C
[tex]{C=163.54\$}[/tex]
To find the cost of materials for the cheapest container, we need to minimize the total cost. The total cost function is C(x) = 20x^2 + 180x. However, there is no minimum cost for the container since it cannot have a negative width, resulting in a cost of $0.
Explanation:Let's denote the width of the rectangular storage container as x. According to the given information, the length of the base is twice the width, so the length would be 2x. The height can be calculated by dividing the volume of the container by the area of the base. Therefore, the height would be 10/(x * 2x) = 5/(2x).
The cost of the base would be the area of the base multiplied by the cost per square meter, which is 10 * (x * 2x) = 20x^2. The cost of the sides would be the sum of each side multiplied by the cost per square meter, which is 6 * (2x * 5) + 6 * (x * 5) + 6 * (2x * 5) = 180x.
To find the cost of materials for the cheapest container, we need to minimize the total cost, which is the sum of the cost of the base and the cost of the sides. Therefore, the total cost function is C(x) = 20x^2 + 180x.
To find the minimum cost, we can take the derivative of C(x) with respect to x, set it equal to 0, and solve for x. The value of x that satisfies this equation will give us the width of the container that minimizes the cost.
C'(x) = 40x + 180 = 0
40x = -180
x = -4.5
Since the width cannot be negative, we disregard this solution.
Therefore, there is no minimum cost for the container since it cannot have a negative width. In this case, the cost of materials for the cheapest container would be $0.
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The following graph is the result of applying a sequence of transformations to the graph of one of the six basic functions. Identify the basic function and write an equation for the given graph.
Answer:
The basic quadratic function is f(x) = x²
The equation of the graph is y = (x - 3)² - 1
Step-by-step explanation:
* Lets explain how to solve the problem
- The graph is a parabola which oped upward
∵ The function is represented by a parabola
∴ The graph is a quadratic function
∴ The basic quadratic function is f(x) = x²
- The vertex of the basic quadratic function is (0 , 0)
∵ From the graph the vertex of the parabola is (3 , -1)
∵ The x coordinate of the basic function change from 0 to 3
∴ The basic function translate 3 units to the right
- If the function f(x) translated horizontally to the right by h units,
then the new function g(x) = f(x - h)
∵ f(x) = x²
∴ The new function g(x) = (x - 3)²
∵ The y-coordinate of the basic function change from 0 to -1
∴ The basic function translate 1 unit down
- If the function f(x) translated vertically down by k units, then the
new function g(x) = f(x) - k
∵ g(x) = (x - 3)²
∴ The new function h(x) = (x - 3)² - 1
∵ h(x) = y
∴ The equation of the graph is y = (x - 3)² - 1
# Note: you can write the equation in general form by solve the
bracket of power 2
∵ (x - 3)² - 1 = (x)(x) - (2)(3)(x) + (3)(3) - 1 = x² - 6x + 9 - 1 = x² - 6x + 8
∴ y = x² - 6x + 8
find the sum of 23+24+25+...+103
Let
[tex]S=23+24+25+\cdots+101+102+103[/tex]
This sum has ___ terms. Its terms form an arithmetic progression starting at 23 with common difference between terms of 1, so that the [tex]n[/tex]-th term is given by the sequence [tex]23+(n-1)\cdot1=22+n[/tex]. The last term is 103, so there are
[tex]103=22+n\implies n=81[/tex]
terms in the sequence.
Now, we also have
[tex]S=103+102+101+\cdots+25+24+23[/tex]
so that adding these two ordered sums together gives
[tex]2S=(23+103)+(24+102)+\cdots+(102+24)+(103+23)[/tex]
[tex]\implies2S=\underbrace{126+126+\cdots+126+126}_{81\text{ times}}=81\cdot126[/tex]
[tex]\implies S=\dfrac{81\cdot126}2\implies\boxed{S=5103}[/tex]
What is the sale price if a down comforter was originally priced to sell at $280 and was reduced by 65%?
Answer:
98
Step-by-step explanation:
100-65=35
Turn 35 into a percent then a decimal, later multiply it by 280
280 X .35=98
Final answer:
The sale price of the down comforter after a 65% discount is $98. This is found by calculating the discount amount of $182 from the original price of $280, then subtracting it to find the final sale price.
Explanation:
To calculate the sale price of the down comforter after a reduction of 65%, we first find the discount amount and then subtract it from the original price. Here's how:
Original price of the down comforter: $280Discount rate: 65%To calculate the discount amount:
Multiply the original price by the discount rate: $280 x 0.65 = $182.Now, subtract the discount amount from the original price to get the sale price: $280 - $182 = $98.The sale price of the down comforter after a 65% discount is $98.
A family has three children. Assuming a boy is as likely as a girl to have been born, what are the following probabilities? Two are girls and one is a boy. Incorrect: Your answer is incorrect. At least 1 is a boy. Incorrect: Your answer is incorrect.
Answer:
3/8 , 3/8
Step-by-step explanation:
Assumption: A boy is as likely as a girl
hence P(B)= P(G)= 1/2= 0.5
family has 3 children
find the probability of
A) two girls and a boy.
it can happen in following way
BGG, GGB or GBG
P(BGG)= P(GGB)= P(GBG) = [tex]\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}[/tex]
= 1/8
therefore , probability of two girl and a boy= [tex]\frac{1}{8} +\frac{1}{8} +\frac{1}{8}[/tex] = 3/8
B) At least One boy
it can happen is in
BGG, BBG, BBB
P(BGG)= P(BBG)= P(BBB) = [tex]\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}[/tex]
= 1/8
therefore , probability of at least one boy= [tex]\frac{1}{8} +\frac{1}{8} +\frac{1}{8}[/tex] = 3/8
A client has an order for 500 mL of NS over 3 hours. The drop factor is 15 gtt/mL. How many gtt/min should be given?
Answer:
42 gtt/min
Step-by-step explanation:
Amount of fluid to be infused = 500 mL
Time = 3 hours = 3×60 = 180 minutes
Tubing drop factor/mL = 15 gtt/mL
Fussion rate = (Amount of fluid to be infused / time in minutes)
Fussion rate = 500/180 = 2.78 mL/min
gtt/min = Tubing drop factor/mL× Fusion rate
⇒gtt/min = 15×(500/180)
⇒gtt/min = 15×(25/9)
⇒gtt/min = 125/3
⇒gtt/min = 41.67
⇒gtt/min = 42
∴42 drops/min (gtt/min) should be given.
4. Suppose you deposit $100 in a savings account that compounds annually at 2%. After 1 year at this rate, the bank changes its rates of compounding to 1.5% annually. Assuming the compounding rate does not change for 4 additional years, how much will your account be worth at the end of the 5 year period?
Answer:
The Final amount in the account after 5 years will be $108.26
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Equations.
Since we are talking about compounding interest we can use the Exponential Growth Formula to calculate the interest over the next couple of years. The formula is the following,
[tex]y = a*(1+r)^{t}[/tex]
Where:
y is the total amount after a given timea is the initial amountr is the interest rate in decimalst is the given timeWe first need to calculate the 2% interest for the first year,
[tex]y = 100*(1+0.02)^{1}[/tex]
[tex]y = 100*1.02[/tex]
[tex]y = 102[/tex]
So after the first year the account will have $102. Now we can use the $102 to calculate the next 4 years of interest using the new interest rate of 1.5%
[tex]y = 102*(1+0.015)^{4}[/tex]
[tex]y = 102*(1.015)^{4}[/tex]
[tex]y = 102*1.0614[/tex]
[tex]y = 108.26[/tex]
The Final amount in the account after 5 years will be $108.26
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In multiple regression analysis, residual analysis is used to test the requirement that ___________. The number of independent variables included in the analysis is correct The variation in the residuals is the same for all predicted values of Y The independent variables are the direct cause of the dependent variable The prediction error is minimized
Answer: The predicted error is minimized.
Step-by-step explanation:
Ideally, residual analysis is used in a linear regression model to measure the appropriateness of the model by examining the residual plots on the graph.
And, residual referred as a difference between the noticed value of the dependent variable (y) and the estimated value (ŷ).
Residual = Noticed value - Estimated value
e = y - ŷ
Multiple regression analysis is used to make a linear model capable of giving predicting an output variable using two or more independent variables. Analysis of the residual is used to to test if the variation in the residuals is the same for all predicted values of y.
Residual values gives the difference between the actual and predicted value of a model. Residual analysis in linear regression is used to test the appropriateness of a linear model for a given data set. Since, the number of independent variables in multiple regression exceeds 1 ; then variation in the predicted values are analysed using the result of the residuals.Therefore, residual analysis in multiple regression tests the variation in the residuals is the same for all predicted values of y.
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A committee has 6 members who have decided on how many votes each should get, but not on the quota to be used for the system. The system so far is [q : 6, 4, 4, 3, 2, 1 ].
a. What value of q would represent a simple majority?
b. What value of q would represent a two-thirds majority?
Answer with explanation:
→Total number of people in the committee=6
→Number of votes in the system represented by q ,
=6+ 4+4+ 3+ 2+ 1=20
a.→Value of q would represent a simple majority if , q>10 that is it would be greater than 50% of votes, means 11 votes.
b.→Value of q that would represent two-thirds majority
[tex]=\frac{2}{3}\times 20\\\\=\frac{40}{3}\\\\=13.33[/tex]
Approximately 14 votes.
Formulate the situation as a system of inequalities. (Let x represent the number of dinghies the company can manufacture and y represent the number of rowboats.)
A boat company manufactures aluminum dinghies and rowboats. The hours of metal work and painting needed for each are shown in the table, together with the hours of skilled labor available for each task. How many of each kind of boat can the company manufacture?
(hours) Dinghy Rowboat Labor Available
Metal Work 2 3 120
Painting 2 2 90
leftbrace6.gif
(labor for metal work)
(labor for painting)
x ? 0, y ? 0
Sketch the feasible region.
Find the vertices. (Order your answers from smallest to largest x, then from smallest to largest y.)
(x, y) =
leftparen1.gif
rightparen1.gif
(x, y) =
leftparen1.gif
rightparen1.gif
(x, y) =
leftparen1.gif
rightparen1.gif
(x, y) =
leftparen1.gif
rightparen1.gif
Answer:
x (smallest to largest) = 0,45 ,55
y (smallest to largest) = 0,10,40
Step-by-step explanation:
(hours) Dinghy Rowboat Labor Available
Metal Work 2 3 120
Painting 2 2 110
Let x represent the number of dinghies the company can manufacture and y represent the number of rowboats.
So, total hours for metal work = [tex]2x+3y[/tex]
So, total hours for Painting = [tex]2x+2y[/tex]
So, equation becomes:
[tex]2x+3y\leq 120[/tex]
[tex]2x+2y\leq 110[/tex]
[tex]x\geq 0[/tex]
[tex]y\geq 0[/tex]
Plot the inequalities
Refer the attached figure
So, the vertices of the feasible region are (0,40),(45,10) and (55,0)
So, x values are 0 , 45 and 55
x represents the number of dinghies
So, x (smallest to largest) = 0,45 ,55
y values are 40,10,0
y represent the number of rowboats.
So, y (smallest to largest) = 0,10,40