Answer:
B, E, F
Step-by-step explanation:
Simply write the terms out and match them.
Final answer:
The correct ordered pairs formed from combining a term in Pattern A with its corresponding term in Pattern B are:
(1, 10), (5, 6), and (7, 4), which are Options B, E, and F, respectively.
Explanation:
The question involves creating ordered pairs from two different number patterns, Pattern A and Pattern B. To find the correct ordered pairs, we match each term in Pattern A with its corresponding term in Pattern B.
Pattern A: 1, 3, 5, 7, 9Pattern B: 10, 8, 6, 4, 2To identify the corresponding terms, we look at the positions: the first term in Pattern A pairs with the first in Pattern B, the second with the second, and so on. Here are the resulting pairs:
(1, 10)(3, 8)(5, 6)(7, 4)(9, 2)Based on the above matching, we evaluate the given options:
Option A: (1, 3) - Incorrect because the first term from Pattern A should be paired with the first term from Pattern B, which is 10.Option B: (1, 10) - Correct as it matches the first term in Pattern A with the first term in Pattern B.Option C: (3, 6) - Incorrect because the second term in Pattern A is 3, but the corresponding second term in Pattern B is 8, not 6.Option D: (5, 4) - Incorrect because the third term in Pattern A is 5, and the corresponding third term in Pattern B is 6, not 4.Option E: (5, 6) - Correct as it matches the third term in Pattern A with the third term in Pattern B.Option F: (7, 4) - Correct as it matches the fourth term in Pattern A with the fourth term in Pattern B.Therefore, the correct ordered pairs when combining a term from Pattern A with its corresponding term from Pattern B are Option B, Option E, and Option F.
By recognizing the series below as a Taylor series evaluated at a particular value of x, find the exact sum of the convergent series. 1 + 3/1! + 9/2! + 27/3! + 81/4! + ... + 3n/n! +.......
Answer:
[tex]e^3[/tex]
Step-by-step explanation:
Given is a series as
[tex]1+\frac{3}{1!} +\frac{3^2}{2!} +...+\frac{3^n}{n!} +...[/tex]
Recall the expansion of
[tex]e^x = 1+x+\frac{x^2}{2!} +...+\frac{x^n}{n!} +...[/tex]
This expansion is valid for all real values of x.
Comparing this with our series we find that x =3
Hence the given series =[tex]e^3[/tex]
Thus we find that the given series can be recognized with the expansion of exponential series with powers of e and here we see that power of e is 3.
So the given Taylor series is equivalent to
[tex]e^3[/tex]
The series in the question is a Taylor series representing e^3x. At x=1, the sum of the series is exactly e^3.
Explanation:The series you provided can be recognized as a Taylor series, an infinite sum of terms calculated from the values of a function's derivatives at a single point. Specifically, it resembles the Taylor series representation of the exponential function ex, which is 1 + x/1! + x2/2! + x3/3! + ... and so on.
Looking at your series 1 + 3/1! + 9/2! + 27/3! + 81/4! ..., we can see that each term 3n/n! is equivalent to (3n)/n!, which can be rewritten as (3^n)(1/n!). This yields a series in the form of 1 + 3x/1! + (3x)2/2! + (3x)3/3! + ... It's apparent that your series can be rewritten as e3x. So when x = 1, the sum of the series is e3, which is exactly the number e cubed.
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Why is the answer E?
Answer:
E
Step-by-step explanation:
For a function to be differentiable at a point, it must be continuous at that point [ f(x⁻) = f(x⁺) ], and smooth at that point [ f'(x⁻) = f'(x⁺) ].
f(-1⁻) = 3(-1) + 5 = 2
f(-1⁺) = -(-1)² + 3 = 2
So the function is continuous.
f'(-1⁻) = 3
f'(-1⁺) = -2(-1) = 2
So the function is not smooth.
Therefore, the derivative f'(-1) does not exist.
At a constant rate of flow, it takes 20 minutes to fill a swimming pool if a large hose is used and 30 minutes if a small hose is used. At these constant rates, how many minutes will it take to fill the pool when both hoses are used simultaneously?
Answer: 12 minutes
Step-by-step explanation:
This is a standard Work Formula question (2 or more 'entities' that work on a task together). When there are just 2 entities and there are no 'twists' to the question, we can use the Work Formula to get to the correct answer.
Work = (A)(B)/(A+B) where A and B are the individual times needed to complete the task.
We're told that two hoses take 20 minutes and 30 minutes, respectively, to fill a pool. We're asked how long it takes the two hoses, working together, to fill the pool.
(20)(30)/(20+30) = 600/50 = 12 minutes to fill the pool.
Simplify
(4x−3+2x2)+(2x+1)
a.8x2−2
b.2x2+2x+4
c.2x2+6x−2
d.2x2−6x−2
Aa. 8x2 - 2
i
Step-by-step explanation:
4x2 = 8
8-3 = 5
5 + 4 =
Answer:
c. 2x2+6x−2
Step-by-step explanation:
Eduardo, Sarah, Maria, Jim, and Tyrone have all been invited to a dinner party. They arrive randomly and each person arrives at a different time.
a. In how many ways can they arrive?
b. In how many ways can Eduardo arrive first and Tyrone last?
c. Find the probability that Eduardo will arrive first and Tyrone last.
There are 120 ways for the five individuals to arrive at the dinner party. If Eduardo always arrives first and Tyrone always arrives last, there are 6 possible orderings. The chance of this specific circumstance happening is 5%.
Explanation:The subject of this question is Permutations and Probability in mathematics.
a. In how many ways can they arrive?
Since there are 5 people and each can arrive at different times, the number of ways they can arrive is equal to the number of permutations of 5 distinct items. This can be calculated as 5 factorial (5!) which equals 5 * 4 * 3 * 2 * 1 = 120. Thus, there are 120 different ways they can arrive.
b. In how many ways can Eduardo arrive first and Tyrone last?
If Eduardo arrives first and Tyrone arrives last, this means there are 3 people left (Sarah, Maria, and Jim) who can arrive in any order in the middle. The number of permutations for these 3 is 3 factorial (3!) which equals 3 * 2 * 1 = 6. Thus, there are 6 different ways Eduardo can arrive first and Tyrone last.
c. Find the probability that Eduardo will arrive first and Tyrone will arrive last.
Probability is calculated by dividing the number of favorable outcomes by the total number of outcomes. From part b, we know that there are 6 favorable outcomes (Eduardo first, Tyrone last). From part a, we know there are 120 total outcomes. Thus, the probability is 6 / 120 = 0.05 or 5%.
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In a study of speed dating, male subjects were asked to rate the attractiveness of their female dates, and a sample of the results is listed below (1equalsnot attractive; 10equalsextremely attractive). Construct a confidence interval using a 99% confidence level. What do the results tell about the mean attractiveness ratings of the population of all adult females? 6, 9, 3, 9, 6, 6, 7, 7, 8, 9, 3, 8
Answer:
The 99% confidence interval is be given by (4.872;8.628)
Step-by-step explanation:
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The dataset is:
6, 9, 3, 9, 6, 6, 7, 7, 8, 9, 3, 8
2) Compute the sample mean and sample standard deviation.
In order to calculate the mean and the sample deviation we need to have on mind the following formulas:
[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex]
The value obtained is [tex]\bar X=6.75[/tex]
[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex]
The sample deviation obtained is [tex]s=2.094[/tex]
3) Find the critical value t* Use the formula for a CI to find upper and lower endpoints
In order to find the critical value we need to take in count that our sample size n =12 <30 and on this case we don't know about the population standard deviation, so on this case we need to use the t distribution. Since our interval is at 99% of confidence, our significance level would be given by [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2 =0.005[/tex]. The degrees of freedom are given by:
[tex]df=n-1=12-1=11[/tex]
We can find the critical values in excel using the following formulas:
"=T.INV(0.005,11)" for [tex]t_{\alpha/2}=-3.106[/tex]
"=T.INV(1-0.005,11)" for [tex]t_{1-\alpha/2}=3.106[/tex]
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]
The next step would be calculate the limits for the interval
Lower interval :
[tex]\bar X - t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]
[tex]6.75 - 3.106 \frac{2.094}{\sqrt{12}}=4.872[/tex]
Upper interval :
[tex]6.75 + 3.106 \frac{2.094}{\sqrt{12}}=8.628[/tex]
So the 99% confidence interval would be given by (4.872;8.628)
99% of the time, when we calculate a confidence interval with a sample of n=12, the true mean of rate of attractiveness of their female dates will be between the 4.872 and 8.628.
A bird is sitting on top of a ten foot pole looking at a piece of bread that is 30 feet from the base of the pole. What is the angle of depression from the bird to the piece of bread.
Answer: The angle of depression is 18.3 degrees
Step-by-step explanation:
The scenario is illustrated in the attached photo
A right angle triangle ABC is formed. The angle of depression, theta forms an alternate angle of # on the ground. To determine the angle, #, we would apply trigonometric ratio
Tan# = opposite side / adjacent side
Opposite side = BC = 10 feets
Adjacent side = AB = 30 feets
Tan # = 10/30 = 0.33
# = tan^-1(0.33)
# = 18.2629
Approximately 18.3 degrees
Since # is alternate to the angle of depression, it means that they are equal. So
The angle of depression is 18.3 degrees
Ali and Renu are buying concert tickets from a web site. There is an 18% 18 % service fee for every ticket bought from the site. If the cost of 2 2 tickets, including the service fee, was $59 $ 59 , what was the cost of each ticket before applying the service fee?
$ 25 was the cost of each ticket before applying the service fee.
Step-by-step explanation:
Given that the cost of two tickets including the service fee = $59
So, the cost of one ticket including the service fee = [tex]\frac{59}{2} = \$ 29.5[/tex]
Service fee on each ticket = 18%
Let the cost of 1 ticket without including the service fee = x
So, according to the data given in the question,
[tex]x+18 \% \text { of } x=29.5[/tex]
[tex]x+\frac{18}{100} \times x=29.5[/tex]
[tex]x+0.18 x=29.5[/tex]
[tex]1.18 x=29.5[/tex]
[tex]x=\frac{29.5}{1.18}=25[/tex]
Hence, cost of each ticket before applying the service fee = $25
Final answer:
The cost of each concert ticket before the 18% service fee was $25. This was calculated by dividing the total cost for two tickets with the service fee ($59) by the total percentage for two tickets (2.36).
Explanation:
The question asks to find the cost of each concert ticket before an 18% service fee is added, given that the total cost for two tickets including the service fee is $59.
To solve this problem, let's denote the cost of one ticket before the service fee as x. Since there is an 18% service fee per ticket, the total cost for one ticket including the service fee is x + 0.18x = 1.18x. As we have two tickets, their total cost would be 2 × 1.18x = 2.36x.
According to the given information, this total cost equals $59. So, we get the equation 2.36x = $59. To find the value of x, divide both sides of the equation by 2.36:
x = $59 ÷ 2.36 = $25
Therefore, the cost of each ticket before the service fee was $25.
The vertical line passing through the vertex of a parabola is called the
Answer:
Axis of symmetry.
Step-by-step explanation:
We have been given an incomplete statement. We are supposed to complete the given statement.
Given statement: The vertical line passing through the vertex of a parabola is called the ________.
We know that a parabola is symmetric about axis of symmetry . The line passing through the vertex of parabola divides the parabola into two mirror images.
Therefore, the vertical line passing through the vertex of a parabola is called the the axis of symmetry.
Listed below are measured amounts of caffeine? obtained in one can from each of 14 brands. Find the? range, variance, and standard deviation for the given sample data. Include appropriate units in the results [mg per 12oz drink; (mg per 12oz of drink)2; brands2; brands]. Are the statistics representative of the population of all cans of the same 14 brands? consumed?
31
52
35
57
0
32
35
52
46
41
30
41
0
0
Answer:
Range is 57.
Variance is 375.143.
standard deviation is 19.37.
Step-by-step explanation:
Consider the provided information.
Range is the difference between highest and lowest data value.
The highest data value is 57 and lowest is 0.
Thus the range is 57-0=57
Range is 57.
Mean is the sum of data value divided by the number of data value:
[tex]\bar x=\frac{31+52+35+57+0+32+35+52+46+41+30+41+0+0}{14}\approx32.286[/tex]
The variance is the sum of squared deviation from the mean divided by n-1.
[tex]s^2 =\frac{\sum(x_i -\bar x)^2}{n - 1}[/tex]
Substitute the respective values in the above formula we get:
[tex]s^2=\frac{(31 - 32.286)^2 +(52-32.286)^2+ ... + (0 -32.286)^2}{14 - 1}\approx 375.143[/tex]
Hence, the variance is 375.143.
Standard deviation is square root of variance.
standard deviation = [tex]\sqrt{375.143}[/tex]
standard deviation ≈ 19.37
Hence, standard deviation is 19.37.
For all cans consumed, the statistics are not representative of the population because in the calculations each brand is weighted equally. Each of the 14 brands of soda is unlikely to be consumed in the same way.
It is very unlikely that all 14 drinks are consumed equally. So,given data is not representative of population
Julie rides her bike from the sports complex to the school. Then she rides from the school to the mall, and then on to the library. Kyle rides his bike from his house to the mall, and then to the library.
The person that traveled the most distance is Julie.
What is an expression?
An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
From the figure,
Julie:
Total distance covered.
= sports complex to school + school to mall + mall + library
= 2/3 + 2/5 + 1(1/3)
= 2/3 + 2/5 + 4/3
= (10 + 6 + 20)/15
= 36/15
= 12/5
= 2(2/5) miles
= 2.4 miles
Kyle:
Total distance covered.
= house to mall + mall to library
= 4/5 + 1(1/3)
= 4/5 + 4/3
= (12 + 20)/15
= 32/15
= 2(2/15) miles
= 2.13 miles
Thus,
Julie has traveled more distance than Kyle.
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Alexis raises 75.23 for charity.Sue raises 3 times as much as alexia.Manuel raises 85.89.How much money do the three raise for charity in all? Show work
Answer: the amount of money raised by the three for charity in all is 386.81
Step-by-step explanation:
Alexis raises 75.23 for charity.
Sue raises 3 times as much as Alexis. This means that the total amount of money raised by Sue is 3 × 75.23= 225.69
The amount of money that Manuel raises is 85.89.
The amount of money raised by all of them(Alexis, Sue and Manuel) would be sum of the amount of money raised by Alexis + the amount of money raised by Sue +
the amount of money raised by Manuel. This becomes
75.23 + 225.69 + 85.89 = 386.81
Answer: 386.81
Step-by-step explanation:
Alexia raises 75.23 = x
Sue raises 3 times Alexia = 3x
= 3 × 75.23
=225.69
Manuel raises 85.89
Total amount raised = Alexia + sue + Manuel
= 75.23 + 225.69 + 85.89
= 386.81
I really need help understanding this! Please help.
Find the measures of all the numbered angles in the figure below.
angle 1= ____
angle 2= ____
angle 3=____
angle 4 _____
angle 5= _____
angle 6= _____
angle 7=_____
Answer:
Below in bold.
Step-by-step explanation:
Note that the lines q and m are parallel.
m < 1 = 180 - 110 = 70 degrees. ( adjacent angles)
m < 2 = 180 - 115 = 65 degrees ( adjacent angles)
m < 3 = m < 2 = 65 degrees ( lines q and m are parallel and 2 and 3 are corresponding angles).
m < 4 = 180 - 65 = 115 degrees (adjacent to < 3).
m < 5 = 180 - m < 1 - m < 3 = 180 - 70 - 65 = 180 - 135 = 45 degrees. ( angles in a triangle add up to 180 degrees).
m < 6 = m < 5 = 45 degrees ( q and m are parallel - alternate angles).
m < 7 =- m < 6 = 45 degrees ( opposite angles).
Anna is an avid reader. Her generous grandparents gave her money for her birthday, and she decided to spend at most $150.00 on books. Reading Spot is running a special: all paperback books are $8.00 and hardback books are $12.00. Anna wants to purchase at least 12 books.
1.) Write a system of inequalities to reach po represent the situation.
2.) Graph the region of the solutions to the inequality.
3.) Name two different solutions for Anna's situation.
Answer:
The solutions for 3 questions are explained one after the other below.
Step-by-step explanation:
1).Let x be the number of paperback books that she buys, y be the number of hardback books that she buys.
for the first condition, i.e, she has decided to spend at most $150.00 on books,the required inequality will be :
[tex]8x+12y\leq 150[/tex]
for the second condition , i.e, she wants to purchase at least 12 books,
the required inequality will be:
[tex]x+y\geq 12[/tex]
2). the graph is in the attachment..
3). x,y are the two required solutions. where,
x =number of paperback books she buys.
y=number of hardback books she buys.
Answer:
1) equations 1, 2, 3 and 4
2) see picture attached (the region of the solutions is in yellow)
3) x = 18.75 and y =0
y = 12.5 and x =0
Step-by-step explanation:
Let's call x the number of paperback books bought and y the number of hardback books bought.
She decided to spend at most $150.00. All paperback books are $8.00 and all hardback books are $12.00. Combining this information we get:
x*8 + y*12 ≤ 150 (eq. 1)
Anna wants to purchase at least 12 books. Mathematically:
x + y ≥ 12 (eq. 2)
On the other hand, both the number of paperback books bought and the number of hardback books bought must be positive, that is:
x ≥ 0 (eq. 3)
y ≥ 0 (eq. 4)
3) One possible solution is got if we make y = 0 and to take eq. 1 as an equality, then:
From eq. 1: x*8 = 150
x = 150/8 = 18.75
equations 2 and 3 are also satisfied
Another option is to make x = 0 and to take eq. 1 as an equality, then:
From eq. 1: y*12 = 150
y = 150/12 = 12.5
equations 2 and 4 are also satisfied
Solve the equation by first using a Sum-to-Product Formula. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate.) sin(5θ) − sin(3θ) = cos(4θ)
Answer:
Solutions of the equation are 22.5°, 30°.
Step-by-step explanation:
The given equation is sin(5θ) - sin(3θ) = cos(4θ)
We take left side of the equation
sin(5θ) - sin(3θ) = [tex]2cos(\frac{5\theta+3\theta}{2})sin(\frac{5\theta-3\theta}{2})[/tex]
= [tex]2cos(4\theta)sin(\theta)[/tex] [From sum-product identity]
Now we can write the equation as
2cos(4θ)sin(θ) = cos(4θ)
2cos(4θ)sinθ - cos(4θ) = 0
cos(4θ)[2sinθ - 1] = 0
cos(4θ) = 0
4θ = 90°
θ = [tex]\frac{90}{4}[/tex]
θ = 22.5°
and (2sinθ - 1) = 0
sinθ = [tex]\frac{1}{2}[/tex]
θ = 30°
Therefore, solutions of the equation are 22.5°, 30°
find the range of the given function f(x)=(x-1)^2+1
a.) [1, infinity)
b.) (- infinity, infinity)
c.) [0, infinity)
d.) (- infinity, 1)
Answer:
a.) [1, infinity)
Step-by-step explanation:
The equation is that of a parabola that opens upward with vertex (1, 1). Hence the minimum value of f(x) is 1, and all values greater than that are part of the range: [1, ∞).
_____
The "vertex form" of the equation of a parabola is ...
f(x) = a(x -h)^2 + k
The vertex is at (h, k). When a > 0, the parabola opens upward. When a < 0, the parabola opens downward. Whichever way it opens, the value k is an extreme value and the limit of the range.
If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli's Law gives the volume V of water remaining in the tank after t minutes as V=5000(1−t40)20≤t≤40. Find the rate at which water is draining from the tank after the following amount of time. (Remember that the rate must be negative because the amount of water in the tank is decreasing.)
Answer:
V'(t) = [tex]-250(1 - \frac{1}{40}t)[/tex]
If we know the time, we can plug in the value for "t" in the above derivative and find how much water drained for the given point of t.
Step-by-step explanation:
Given:
V = [tex]5000(1 - \frac{1}{40}t )^2[/tex] , where 0≤t≤40.
Here we have to find the derivative with respect to "t"
We have to use the chain rule to find the derivative.
V'(t) = [tex]2(5000)(1 - \frac{1}{40} t)d/dt (1 - \frac{1}{40}t )[/tex]
V'(t) = [tex]2(5000)(1 - \frac{1}{40} t)(-\frac{1}{40} )[/tex]
When we simplify the above, we get
V'(t) = [tex]-250(1 - \frac{1}{40}t)[/tex]
If we know the time, we can plug in the value for "t" and find how much water drained for the given point of t.
A farmer needs to enclose three sides of a field with a fence (the fourth side is a river). The farmer has 49 yards of fence and wants the field to have an area of 294 sq-yards. What should the dimensions of the field be? (For the purpose of this problem, the width will be the smaller dimension (needing two sides); the length with be the longer dimension (needing one side). Additionally, the length should be as long as possible.)
Answer:
The length of the field = 24.5 yards
The width of the field = 12 yards.
Step-by-step explanation:
If "w" is the width, then the length is 49 - 2w.
The area of the rectangle field = length × width
= w(49 - 2w)
Area = [tex]49w - 2w^2[/tex]
This a quadratic equation, the vertex of x coordinate is w
w = [tex]\frac{-b}{2a}[/tex]
Here a = -2 and b = 49
w = [tex]\frac{-49}{2(-2)} = \frac{-49}{-4} = 12.25[/tex].
So width of the field is 12.25 yards.
The length of the filed = 49 - 2(12.25) = 24.5
Area = 12.25 × 24.5 = 300.125 square yards.
The field has an area of 294.
Therefore, the length must be 24.5 yards and width must be 12 yards.
24.5 × 12 =294 square yards.
Therefore, the length of the field = 24.5 yards
the width of the field = 12 yards.
To find the dimensions of the field, we can use equations based on the area and perimeter. We set up equations for the area and perimeter and solve them simultaneously to find the values of width and length.
Explanation:To find the dimensions of the field, we can use the formula for finding the area of a rectangle: length x width. Let's assume the length of the field is x yards. Since the width is the smaller dimension and requires two sides, we'll represent it as 2w yards. Given that the area of the field is 294 sq-yards, we have the equation x * 2w = 294.
Additionally, we know that the farmer has 49 yards of fence, and he needs to enclose three sides of the field. The three sides are two sides of width (2w) and one side of length (x). So, the total length of the fence needed is 2w + 2w + x = 49.
Simplifying the equation, we have 4w + x = 49. Substituting x from the first equation into the second equation, we get 4w + (294 / 2w) = 49.
Now, we can solve this equation to find the value of w. Once we have the value of w, we can substitute it back into the first equation to find the value of x.
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(25 POINTS) PLEASE HELP WILL GIVE BRAINLIEST, THANKS AND 5 STAR RATING!!!
5 QUESTIONS SHOW WORK!
Answer:
Step-by-step explanation:
The goals against average (A) for a professional hockey goalie is determined using the formula A = 60 . In the formula, g represents the number of goals scored against the goalie and t represents the time played, in minutes. Which is an equivalent equation solved for g?
A_At/60 = g
B_ A/60t= g
C_60A/t = g
D_60At = g
Answer:
A. [tex]\frac{At}{60}=g[/tex]
Step-by-step explanation:
Consider the formula [tex]A = 60(\frac{g}{t})[/tex] is given,
We have to find : The formula for g.
For this we need to isolate g in one side of the equation,
Multiply both sides of the equation by t, ( using multiplicative property of equality )
We get,
[tex]At = 60g[/tex]
Divide both sides by 60 ( division property of equality )
We get,
[tex]\frac{At}{60}=g[/tex]
Which is the required equivalent equation.
That is,
OPTION A would be correct.
Answer: A. At/60 = g
Step-by-step explanation: Consider the formula is given,
We have to find : The formula for g.
For this we need to isolate g in one side of the equation,
Multiply both sides of the equation by t, ( using multiplicative property of equality )
We get,
Divide both sides by 60 ( division property of equality )
We get,
Which is the required equivalent equation.
That is,
OPTION A would be correct.
Which are the solutions of x2 = –13x – 4? 0, 13 0, –13 StartFraction 13 minus StartRoot 153 EndRoot Over 2 EndFraction comma StartFraction 13 + StartRoot 153 EndRoot Over 2 EndFraction StartFraction negative 13 minus StartRoot 153 EndRoot Over 2 EndFraction comma StartFraction negative 13 + StartRoot 153 EndRoot Over 2 EndFraction
Answer:
[tex]x_{1}=\frac{-13+\sqrt{153}}{2}\\x_{2}=\frac{-13-\sqrt{153}}{2}[/tex]
Step-by-step explanation:
The given expression is
[tex]x^{2}=-13x-4[/tex]
To solve this quadratic equation, we first need to place all terms in one side of the equation sign
[tex]x^{2} +13x+4=0[/tex]
Now, to find all solutions of this expression, we have to use the quadratic formula
[tex]x_{1,2}=\frac{-b\±\sqrt{b^{2}-4ac}}{2a}[/tex]
Where [tex]a=1[/tex], [tex]b=13[/tex] and [tex]c=4[/tex]
Replacing these values in the formula, we have
[tex]x_{1,2}=\frac{-13\±\sqrt{(13)^{2}-4(1)(4)}}{2(1)}\\x_{1,2}=\frac{-13\±\sqrt{169-16}}{2}=\frac{-13\±\sqrt{153}}{2}[/tex]
So, the solutions are
[tex]x_{1}=\frac{-13+\sqrt{153}}{2}\\x_{2}=\frac{-13-\sqrt{153}}{2}[/tex]
If we approximate each solution, it would be
[tex]x_{1}=\frac{-13+\sqrt{153}}{2}\approx -0.32\\\\x_{2}=\frac{-13-\sqrt{153}}{2} \approx -12.68[/tex]
Answer:
D on Edge
Step-by-step explanation:
What’s the area of the figure?
Answer:
Step-by-step explanation:
area[tex]=\frac{1}{2}*5*8\\=20~yd^2[/tex]
A student dance committee is to be formed consisting of 2 boys and 4 girls. If the membership is to be chosen from 5 boys and 6 girls, how many different committees are possible?
150 different committees are possible
Solution:Given that a student dance committee is to be formed consisting of 2 boys and 4 girls
The membership is to be chosen from 5 boys and 6 girls
To find : number of different possible committees
A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected
The formula for combination is given as:
[tex]n C_{r}=\frac{n !}{(n-r) ! r !}[/tex]
where "n" represents the total number of items, and "r" represents the number of items being chosen at a time
We have to select 2 boys from 5 boys
So here n = 5 and r = 2
[tex]\begin{aligned} 5 C_{2} &=\frac{5 !}{(5-2) ! 2 !}=\frac{5 !}{3 ! 2 !} \\\\ 5 C_{2} &=\frac{5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 2 \times 1} \\\\ 5 C_{2} &=10 \end{aligned}[/tex]
We have to select 4 girls from 6 girls
Here n = 6 and r = 4
[tex]\begin{aligned} 6 C_{4} &=\frac{6 !}{(6-4) ! 4 !}=\frac{6 !}{2 ! 4 !} \\\\ 6 C_{4} &=\frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 4 \times 3 \times 2 \times 1}=15 \end{aligned}[/tex]
Committee is to be formed consisting of 2 boys and 4 girls:
So we have to multiply [tex]5 C_{2}[/tex] and [tex]6 C_{4}[/tex]
[tex]5 C_{2} \times 6 C_{4}=10 \times 15=150[/tex]
So 150 different committees are possible
Final answer:
The question is a combinatorial problem in mathematics, where the goal is to calculate the number of different committees that can be formed from 5 boys and 6 girls. To solve this, the combinations formula is applied separately to choose 2 boys from 5 and 4 girls from 6, and the results are multiplied.
Explanation:
The question asks about the number of different committees that can be formed from a group of boys and girls. This is a combinatorial problem involving calculations to find the different possible combinations that can be made using a subset of a larger set. To solve this, you would use the combinations formula which is given by C(n, k) = n! / (k!(n-k)!), where n is the total number of items to choose from, k is the number of items to choose, and ! denotes the factorial of a number.
To determine how many different committees are possible, we calculate the number of ways to choose 2 boys from 5, and 4 girls from 6 separately, and then multiply these two results:
The number of ways to choose 2 boys from 5 is C(5, 2)
The number of ways to choose 4 girls from 6 is C(6, 4)
Therefore, the total number of different committees possible is C(5, 2) * C(6, 4).
Which is the graph of y = ⌊x⌋ – 2?
Answer:
The third graph from left to right
Step-by-step explanation:
The function [tex]f(x)=\left [ x \right ][/tex] is called Greatest Integer Function of x is such that it returns the largest integer less than or equal to x
Some examples of points are (0,0),(0.5,0),(1,1),(1.9,1),(-0.7,-1)
Since our function is
[tex]f(x)=\left [ x \right ]-2[/tex]
We must subtract 2 to the points above like
(0,-2),(0.5,-2),(1,-1),(1.9,-1),(-0.7,-3)
The only graph that complies with such requirements is the third one
What are the solutions of the quadratic equation below? -7x2 - 23x + 10 = 0 A. B. C. D.
Answer:
There are no like terms.
Answer:
Step-by-step explanation:
Suppose the cost C(x), to build a football stadium of x thousand square feet is approximated by C(x) = 7,250,000/ x + 60 . How many square feet was the stadium if the cost of the stadium was $8,000?
The football stadium was approximately 5,000 thousand square feet.
Explanation:The given cost function is [tex]\( C(x) = \frac{7,250,000}{x} + 60 \)[/tex], where C(x) represents the cost to build a football stadium of x thousand square feet. To find the size of the stadium when the cost is $8,000, we set C(x) equal to $8,000 and solve for x :
[tex]\[ 8,000 = \frac{7,250,000}{x} + 60 \][/tex]
Subtracting 60 from both sides:
[tex]\[ 7,940 = \frac{7,250,000}{x} \][/tex]
Now, solving for x , we multiply both sides by x :
[tex]\[ x = \frac{7,250,000}{7,940} \][/tex]
Performing the division:
x approx 912.66
Since x represents the size of the stadium in thousand square feet, the final answer is approximately 912.66 thousand square feet. Therefore, the stadium was approximately 5,000 thousand square feet.
In summary, to determine the size of the football stadium, we set the cost function equal to the given cost, solved for x , and found that the stadium was approximately 912.66 thousand square feet.
The football stadium was approximately x = 916.67 thousand square feet.
Explanation:The given cost function for building a football stadium is [tex]\(C(x) = \frac{7,250,000}{x} + 60\)[/tex], where \(x\) represents the size of the stadium in thousand square feet. We are asked to find the size of the stadiumx when the cost[tex](\(C(x)\))[/tex] is $8,000.
To solve for x, we set [tex]\(C(x)\)[/tex] equal to $8,000 and solve for x:
[tex]\[ 8,000 = \frac{7,250,000}{x} + 60 \][/tex]
First, subtract 60 from both sides:
[tex]\[ 7,940 = \frac{7,250,000}{x} \][/tex]
Next, multiply both sides by x to isolate x in the denominator:
[tex]\[ 7,940x = 7,250,000 \][/tex]
Finally, solve for [tex]\(x\)[/tex] by dividing both sides by 7,940:
[tex]\[ x = \frac{7,250,000}{7,940} \approx 916.67 \text{ thousand square feet} \][/tex]
Therefore, the football stadium was approximately 916.67 thousand square feet in size.
In summary, by substituting the given cost into the cost function and solving the resulting equation, we find that the stadium size is approximately 916.67 thousand square feet. This process involves algebraic manipulation to isolate [tex]\(x\)[/tex] and perform the necessary arithmetic calculations.e
Find the area and perimeter of ABC at right. Give approximate (decimal) answers, not exact answers
Answer:
Area of Δ ABC = 21.86 units square
Perimeter of Δ ABC = 24.59 units
Step-by-step explanation:
Given:
In Δ ABC
∠A=45°
∠C=30°
Height of triangle = 4 units.
To find area and perimeter of triangle we need to find the sides of the triangle.
Naming the end point of altitude as 'O'
Given [tex]BO\perp AC[/tex]
For Δ ABO
Since its a right triangle with one angle 45°, it means it is a special 45-45-90 triangle.
The sides of 45-45-90 triangle is given as:
We are given BO (Leg 1) [tex]x=4[/tex]
∴ AO (Leg2) [tex]=x=4[/tex]
∴ AB (hypotenuse) [tex]=x\sqrt2=4\sqrt2=5.66 [/tex]
For Δ CBO
Since its a right triangle with one angle 30°, it means it is a special 30-60-90 triangle.
The sides of 30-60-90 triangle is given as:
We are given BO (side opposite 30° angle) [tex]=x=4[/tex]
CO (side opposite 60° angle) [tex]=x\sqrt3=4\sqrt3=6.93[/tex]
BC (Hypotenuse) [tex]=2x=2\times 4 =8[/tex]
Length of side AC is given as sum of AO and CO
[tex]AC=AO+CO=4+6.93=10.93[/tex]
Perimeter of Δ ABC= Sum of sides of triangle
⇒ AB+BC+AC
⇒ [tex]5.66+8+10.93[/tex]
⇒ [tex]24.59[/tex] units
Area of Δ ABC = [tex]\frac{1}{2}\times base\times height[/tex]
⇒ [tex]\frac{1}{2}\times 10.93\times 4[/tex]
⇒ [tex]21.86[/tex] units square
This is my last problem on this sample work. Dx Please help! There are no multiple choice options so this is a free for all. Thank you!!
Answer:
[tex]\left\{\begin{array}{l}y\ge 2x+4\\ \\y<-x+2\end{array}\right.[/tex]
Step-by-step explanation:
1. The solid line passes trough the points (0,4) and (-2,0). The equation of this line is:
[tex]\dfrac{x-0}{-2-0}=\dfrac{y-4}{0-4}\\ \\y-4=2x\\ \\y=2x+4[/tex]
The origin doesn't belong to the shaded region, so its coordinates do not satisfy the inequality. Thus,
[tex]y\ge 2x+4[/tex]
2. The dotted line passes trough the points (0,2) and (2,0). The equation of this line is:
[tex]\dfrac{x-0}{2-0}=\dfrac{y-2}{0-2}\\ \\y-2=-x\\ \\y=-x+2[/tex]
The origin belongs to the shaded region, so its coordinates satisfy the inequality. Thus,
[tex]y< -x+2[/tex]
Hence, the system of two inequalities is
Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machines Y. AT these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets.
A. 4
B. 6
C. 8
D. 10
E. 12
Answer:
E. 12 days
Step-by-step explanation:
So first, we need to find the rates at which each machine will produce w widgets. For machine y, its rate would be:
[tex]y=\frac{W}{T}[/tex]
where W is the number of widgets produced and T is the time it takes to produce them.
We know that x takes 2 more days to produce the same amount of widgets, so the time it takes machine x to produce them can be written as T+2. This will give us the following rate for machine x:
[tex]x=\frac{W}{T+2}[/tex]
the problem also tells us that the two machines working together will produce 5W/4 widgets in 3 days, so if we add the rates for x and y, we will get the total rate which would be:
[tex]x+y=\frac{5W/4}{3}[/tex]
which can be simplified to:
[tex]x+y=\frac{5W}{12}[/tex]
we can now substitute the rates for x and y in the equation so we get:
[tex]\frac{W}{T+2}+\frac{W}{T}=\frac{5W}{12}[/tex]
we can simplify this equation by dividing everything into W, so we get:
[tex]\frac{1}{T+2}+\frac{1}{T}=\frac{5}{12}[/tex]
and we can multiply everything by the LCD. In this case the LCD is 12T(T+2) so we get:
[tex]\frac{1}{T+2}(12T)(T+2)+\frac{1}{T}(12T)(T+2)=\frac{5}{12}(12T)(T+2)[/tex]
which simplifies to:
12T+12(T+2)=5T(T+2)
we can do the respective multiplications so we get:
[tex]12T+12T+24=5T^{2}+10T[/tex]
which simplifies to:
[tex]24T+24=5T^{2}+10T[/tex]
and now we can set the equation equal to zero so we end up with:
[tex]5T^{2}+10T-24T-24=0[/tex]
which simplifies to:
[tex]5t^{2}+14T-24=0[/tex]
now we can solve this by any of the available methods there are to solve quadratic equations. I will solve it by factoring, so we get:
(5T+6)(T-4)=0
so we can set each of the factors equal to zero so we get:
5T+6=0
[tex]T=-\frac{6}{5}[/tex]
this answer isn't valid because there is no such thing as a negative time. So we find the next time then:
T-4=0
T=4
So it takes 4 days for machine x to produce W widgets. We can now rewrite x's rate like this:
[tex]x=\frac{W}{T+2}[/tex]
so
[tex]x=\frac{W}{4+2}[/tex]
[tex]x=\frac{W}{6}[/tex]
With this information, we know that the number of wigets produced can be found by using the following formula:
W=xd
in this case d is the number of days (this is for us not to confuse the previous T with the new time)
so when solving for d we get that:
[tex]d=\frac{W}{x}[/tex]
so when substituting we get that:
[tex]d=\frac{2W}{W/6}[/tex]
when simplifying we get that:
d=12
On the basis of data collected during an experiment, a biologist found that the growth of a fruit fly population (Drosophila) with a limited food supply could be approximated by
N(t) = 600/1+39e^-0.16t
(a) What was the initial fruit fly population in the experiment? where t denotes the number of days since the beginning of the experiment.
(b) What was the population of the fruit fly colony on the t = 11 day? (Round your answer to the nearest integer.)
Answer:
(a). 15
(b). 78
Step-by-step explanation:
Growth of the population of a fruit fly is modeled by
N(t) = [tex]\frac{600}{1+39e^{-0.16t} }[/tex]
where t = number of days from the beginning of the experiment.
(a). For t = 0 [Initial population]
N(0) = [tex]\frac{600}{1+39e^{-0.16\times 0} }[/tex]
= [tex]\frac{600}{1+39}[/tex]
= [tex]\frac{600}{40}[/tex]
= 15
Initial population of the fruit flies were 15.
(b).Population of the fruit fly colony on 11th day.
N(11) = [tex]\frac{600}{1+39e^{-0.16\times 11} }[/tex]
= [tex]\frac{600}{1+39e^{-1.76} }[/tex]
= [tex]\frac{600}{1+39\times 0.172 }[/tex]
= [tex]\frac{600}{1+6.71}[/tex]
= [tex]\frac{600}{7.71}[/tex]
= 77.82
≈ 78
On 11th day number of fruit flies colony were 78.