[tex]\bf \textit{Sum to Product Identities} \\\\ cos(\alpha)+cos(\beta)=2cos\left(\cfrac{\alpha+\beta}{2}\right)cos\left(\cfrac{\alpha-\beta}{2}\right) \\\\\\ cos(\alpha)-cos(\beta)=-2sin\left(\cfrac{\alpha+\beta}{2}\right)sin\left(\cfrac{\alpha-\beta}{2}\right) \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]
[tex]\bf \cfrac{cos(4\theta )+cos(2\theta )}{cos(4\theta )-cos(2\theta )}\implies \cfrac{2cos\left( \frac{4\theta +2\theta }{2} \right)cos\left( \frac{4\theta -2\theta }{2} \right)}{-2sin\left( \frac{4\theta +2\theta }{2} \right)sin\left( \frac{4\theta -2\theta }{2} \right)} \implies \cfrac{cos\left( \frac{6\theta }{2} \right)cos\left( \frac{2\theta }{2} \right)}{-sin\left( \frac{6\theta }{2} \right)sin\left( \frac{2\theta }{2} \right)}[/tex]
[tex]\bf \cfrac{cos(3\theta )cos(\theta )}{-sin(3\theta )sin(\theta )}\implies -\cfrac{cos(3\theta )}{sin(3\theta )}\cdot \cfrac{cos(\theta )}{sin(\theta )}\implies -cot(3\theta )cot(\theta )[/tex]
The given expression is:
(cos 4θ + cos 2θ) / (cos 4θ - cos 2θ)
To simplify this expression, we can use the formula cot A sin C + cos B cos C = cot A sin B. Applying this formula gives us -cot 3θ cot θ as the simplified form of the expression.
Darren teaches a class of 25 students. He assigns homework 3 times a week, and each assignment consists of 12 problems. How many problems must Darren correct each week?
A.
225 problems
B.
890 problems
C.
900 problems
D.
1,000 problems
Answer:
Option C is correct.
Step-by-step explanation:
Homework is assigned: 3 times a week
Each assignment consists problems = 12
Total questions in 1 week = 12*3 = 36
Total no of students = 25
So, Problems Darren must correct each week = Total no of students * Total questions in 1 week
Problems Darren must correct each week = 25*36
Problems Darren must correct each week = 900
So, Option C is correct.
For this case we have that each week assign 3 tasks, each of 12 problems, then multiply:
[tex]3 * 12 = 36[/tex]
Thus, assign 36 weekly problems to each student. Darren has 25 students, so, multiplying, we have:
[tex]36 * 25 = 900[/tex]
Thus, Darren must correct 900 weekly problems.
Answer:
Option C
Write the equation of the line that is perpendicular to the line 5y=x−5 through the point (-1,0).
A. y= 1/5x+5
B. y= 1/5x−5
C. y= −5x−5
D. y= −5x+5
Steps:
---> Re arrange equation to get the format: y = mx + c
---> Work out the perpendicular gradient from the first equation
----> Substitute the x and y coordinates of point (-1, 0) and the perpendicular gradient into y = mx + c and work out c
---> Finally, substitute the perpendicular gradient and the value for c into y =mx + c to get the gradient of the perpendicular line:
__________________________________________
Rearranging equation into the format: y = mx + c:
[tex]5y = x - 5[/tex] (Just divide both sides by y)
[tex]y = \frac{1}{5}x -1[/tex]
___________________________________________
Working out the perpendicular gradient:
To work out the perpendicular gradient, we just take the negative reciprocal of the gradient of [tex]y = \frac{1}{5}x -1[/tex]
Note: negative reciprocal means we just flip the fraction and put a minus sign.
The regular gradient is: [tex]\frac{1}{5}[/tex]
So the perpendicular gradient is the negative reciprocal of [tex]\frac{1}{5}[/tex]
which is -5 (note: [tex]\frac{-5}{1}[/tex] is just 5-)
___________________________________________
Now lets substitute in the values for the gradient (m), the y coord (0) and x coord (-1) of the point (-1, 0) into y = mx + c, and solve for c:
y = mx + c (substitute in all known values)
0 = -5(-1) + c (the -1 times -5 will make + 5)
0 = 5 + c (subtract 5 from both sides to cancel out the + 5)
-5 = c
so c = -5
____________________________________________
Finally, just substitute in the perpendicular gradient and the value for c into y = mx + c to get the equation of the perpendicular line:
y = mx + c (substitute in the perp. gradient and c)
y = -5x - 5
____________________________________________________
Answer:The equation to the line perpendicular to 5y = x - 5 through point (-1, 0) is :
C. y = -5x - 5
_______________________________________________
A quicker way to get equation of the perpendicular line once you know the perp. gradient is to use the equation:
y - y1 = m (x - x1)
y1 is the y coordinate of (-1, 0)
x1 is the x coordinate of (-1, 0)
m is the perpendicular gradient.
y - y1 = m (x - x1) (Substitute in values)
y - 0 = -5 ( x - - 1) (simplify)
y = -5 (x + 1) (expand the brackets)
y = -5x - 5
The computers of six faculty members in a certain department are to be replaced. Two of the faculty members have selected laptop machines and the other four have chosen desktop machines. Suppose that only two of the setups can be done on a particular day, and the two computers to be set up are randomly selected from the six (implying 15 equally likely outcomes; if the computers are numbered 1, 2,…, 6, then one outcome consists of computers 1 and 2, another consists of computers 1 and 3, and so on). a. What is the probability that both selected setups are for laptop computers
Answer:
The answer is 0.067.
Step-by-step explanation:
Let the entire sample size be = s
Now there are 2 laptops in sample size, hence these can be chosen in one way only.
The required probability that both selected setups are for laptop computers can be found as:
[tex]p(two laptops)=\frac{s(two laptops)}{s}[/tex]
= [tex]\frac{1}{15}[/tex] or 0.067.
So, the probability is 0.067.
The probability of both selected setups being for laptop computers is 2/15.
Explanation:The probability of both selected setups being for laptop computers can be calculated as the ratio of favorable outcomes to total outcomes. Out of the six computers, two have been selected to be laptops. The first laptop can be any of the two laptops, and the second laptop can be any of the remaining one laptop. Therefore, the probability of both selected setups being for laptop computers is 2/15.
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A student is solving the system of equations below. In which line is a mistake first made?
2x + y - 2z = 23
3x + 2y + z = 11
x - y + z = -2
Line 1 z = 11 + 3x + 2y
Line 2 2x + y - 2(11 + 3x + 2y) = 23
Line 3 -4x - 3y = 45
Line 4 x - y + (11 + 3x + 2y) = - 2
Line 5 4x + y = - 13
Line 6 -2y = 32
Line 7 y = 16, x = ¾, z = -11/4
Answer:
1st line
Step-by-step explanation:
Should be z=11-3x-2y, then solve from there.
The Ericsson method is one of several methods claimed to increase the likelihood of a baby girl. In a clinical trial, results could be analyzed with a formal hypothesis test with the alternative hypothesis of pgreater than>0.5, which corresponds to the claim that the method increases the likelihood of having a girl, so that the proportion of girls is >0.5. If you have an interest in establishing the success of the method, which of the following P-values would you prefer: 0.999, 0.5, 0.95, 0.05, 0.01, 0.001? Why?
Answer:
The preferred p-value among the options offered is 0.001. The reason is because p-value represents the minimum probability of committing the type I error, that is, the probability of rejecting the null hypothesis, with the information contained in the sample, given that this hypothesis is true. The p-value of 0.001 is the lowest value proposed for this probability. Therefore, it supports more evidence that the null hypothesis is false.
Step-by-step explanation:
The preferred P-value among the options provided is 0.001.
0.001 is the preferred p-value simply because it shows the minimum probability of committing an error, it also implies that the probability of rejecting the null hypothesis based on the information presented in the sample, considering that the hypothesis is true.
Further ExplanationTherefore, the p-value 0.001 is the lowest value proposed for this probability and it also corresponds to the sample of evidence that supports the alternative hypothesis which shows the method is effective.
Hypothesis testing refers to an act in statistics in which an analyst test the assumption as regards to a population parameter.
The methodology to be used by analysts is based on the nature of the data that are used and the purpose of the analysis
An analysts test a sample to accept or reject a null hypothesis. The outcome of the tests will reveal the analysis if his primary hypothesis is true or not. If the analysis of the tests is not true, it then means the analysts will have to formulate a new hypothesis and do the analysis again.
It implies the analysts will continue to repeat the process until the analysis shows the data hypothesis is true.
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hypothesis testericsson methodanalysisproportionprobabilityThe percent body fat in a random sample of 36 men aged 20 to 29 has a sample mean of 14.42. Find a 99% confidence interval for the mean percent body fat in all men aged 20 to 29. Assume that percent body fat follows a Normal distribution, with a standard deviation of 6.95.
A) (0.8, 28.04)B) (12.15, 16.69)C) (12.51, 16.33)D) (12.07, 16.77)
Answer: [tex](11.44,\ 17.4)[/tex]
Step-by-step explanation:
The confidence interval for population mean is given by :-
[tex]\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}[/tex]
Given : Sample size : [tex]n=36[/tex]
Sample mean : [tex]\ovreline{x}=14.42[/tex]
Standard deviation : [tex]\sigma=6.95[/tex]
Significance level : [tex]\alpha=1-0.99=0.01[/tex]
Critical value : [tex]z_{\alpha/2}=z_{0.005}=2.576[/tex]
Now, the 99% confidence interval for the mean percent body fat in all men aged 20 to 29 will be :-
[tex]14.42\pm (2.576)\dfrac{6.95}{\sqrt{36}}\\\\\approx14.42\pm2.98\\\\=(14.42-2.98,\ 14.42+2.98)=(11.44,\ 17.4)[/tex]
Function f approximately represents the trajectory of an airplane in an air show, where x is the horizontal distance of the flight, in kilometers.
f(x)=88x^2-264x+300
Determine the symmetry of the function.
A. The trajectory of the airplane is symmetric about the line x = 102 km, which indicates that the height of the airplane when it moves a horizontal distance of 101 km is the same as the height of the airplane when it moves a horizontal distance of 103 km.
B. The trajectory of the airplane is not symmetric.
C. The trajectory of the airplane is symmetric about the line x = 1.5 km, which indicates that the height of the airplane when it moves a horizontal distance of 0.5 km is the same as the height of the airplane when it moves a horizontal distance of 2.5 km.
D. The trajectory of the airplane is symmetric about the line x = 2 km, which indicates that the height of the airplane when it moves a horizontal distance of 1 km is the same as the height of the airplane when it moves a horizontal distance of 3 km.
Answer:
C. is your answer
Step-by-step explanation:
In order to determine the line of symmetry, it would help to put this standard form parabola into vertex form, which is
[tex]y=a(x-h)^2+k[/tex],
where x = h is the equation of the line of symmetry.
To get this into vertex form we will complete the square. The first couple of steps I will combine into 1. We will set the quadratic equal to zero, then move the constant over to the other side:
[tex]88x^2-264x=-300[/tex]
The next rule is that the leading coefficient HAS to be a positive 1. Ours is a positive 88, so we have to factor it out:
[tex]88(x^2-3x)=300[/tex]
Now we can perform the process of completing the square. The rule is to take half the linear term, square it, and add it to both sides. Our linear term is 3. Half of 3 is 3/2, and 3/2 squared is 9/2. We will add 9/2 inside the parenthesis on the left, but don't forget about that 88 sitting out front which refuses to be ignored. It serves as a multiplier into the parenthesis. What we actually added in, then, was 88(9/2) which is 198:
[tex]88(x^2-3x+\frac{9}{4})=-300+198[/tex]
The purpose of completing the square is to give us a perfect square binomial which serves as the axis of symmetry of the parabola and also gives us the h coordinate of the vertex. We will state that binomial and at the same time do the addition on the right:
[tex]88(x-\frac{3}{2})^2=-102[/tex]
Now we can move the constant back over and set it back equal to y:
[tex]y=88(x-\frac{3}{2})^2+102[/tex]
From that form you can see that the equation of the line of symmetry is x = 1.5. The coordinates of the vertex are (1.5, 102). If we move 1 unit to the left of the vertex, or 1 unit to the right of the vertex, we will be at the same height.
C then is your answer.
The solution is : C. is the answer.
C. The trajectory of the airplane is symmetric about the line x = 1.5 km, which indicates that the height of the airplane when it moves a horizontal distance of 0.5 km is the same as the height of the airplane when it moves a horizontal distance of 2.5 km.
What is parabola?The parabola is a plane curve which is mirror symmetrical and is approximately U-shaped. It fits several superficial different mathematical descriptions.
here, we have,
In order to determine the line of symmetry, it would help to put this standard form parabola into vertex form, which is
y = a (x-h)^2 + k
,where x = h is the equation of the line of symmetry.
To get this into vertex form we will complete the square. The first couple of steps I will combine into 1. We will set the quadratic equal to zero, then move the constant over to the other side:
88x^2 - 264x = -300
The next rule is that the leading coefficient HAS to be a positive 1. Ours is a positive 88, so we have to factor it out:
88( x^2 - 3x) = 300
Now we can perform the process of completing the square. The rule is to take half the linear term, square it, and add it to both sides. Our linear term is 3. Half of 3 is 3/2, and 3/2 squared is 9/2. We will add 9/2 inside the parenthesis on the left, but don't forget about that 88 sitting out front which refuses to be ignored. It serves as a multiplier into the parenthesis. What we actually added in, then, was 88(9/2) which is 198:
88( x^2 - 3x + 9/4 ) = -300 + 198
The purpose of completing the square is to give us a perfect square binomial which serves as the axis of symmetry of the parabola and also gives us the h coordinate of the vertex. We will state that binomial and at the same time do the addition on the right:
88( x - 3/2)^2 = -102
Now we can move the constant back over and set it back equal to y:
y = 88( x - 3/2)^2 + 102
From that form you can see that the equation of the line of symmetry is x = 1.5. The coordinates of the vertex are (1.5, 102). If we move 1 unit to the left of the vertex, or 1 unit to the right of the vertex, we will be at the same height.
C then is the answer.
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What is the coefficient of x^4y^4 in the expansion (x+y)^8?
A. 1
B. 40
C. 70
D. It does not exist
Answer:
C. 70
Step-by-step explanation:
In the expansion of (a + b)^n, the k-th term is ...
nCk·a^(n-k)b^k . . . . . k = 0 to n; nCk = n!/(k!(n-k)!)
Here, we have n=8, k=4, so the term of interest is ...
8C4·x^4y^4 = (8·7·6·5)/(4·3·2·1)x^4y^4 = 70x^4y^4
The coefficient of the term is 70.
What kind of quadrilateral is the shape shown here?
The matching arrow labels indicate that two opposite sides are parallel.
Answer:
C is correct as well as D.
Step-by-step explanation:
The rhombus also shares the common traits that the picture shows
For a certain game, three numbers are drawn, each from 1 to 10. A player can win if the product of his or her three numbers is odd. Initially, the same number may be picked up to three times. Later, the rules are changed so that each number may only be picked once. Approximately how much does this reduce a person's probability of winning?
Answer:
Step-by-step explanation:
if you can pick the same card 3 times the probability of winning is
[tex](\frac{5}{10})^{3} = \frac{1}{8}[/tex]
but if you remove each card after you've picked them it's :
[tex]\frac{5}{10}.\frac{4}{9}.\frac{3}{8}= \frac{60}{720}[/tex]
notice if you remove an odd card every time you pick one of them you are also removing one of the overall cards
and if you subtract these two you get : [tex]\frac{1}{8} - \frac{60}{720} =0.0417[/tex]
which is like 4 percent
NASA received three messages in a strange language from a distant planet. Scientists studied these messages and found that "Floos Beiling Hoomp" means "We come in peace" and "Moog Naline Floos" means "Peace, Love, and Freedom " and "Beiling Boog Vladi" means "In the last hour". What does "Hoomp" mean?
Answer:
Hoomp means "we come".
Step-by-step explanation:
"Floos Beiling Hoomp" means "We come in peace"
"Moog Naline Floos" means "Peace, Love, and Freedom "
"Beiling Boog Vladi" means "In the last hour"
Comparing 1st two statements, we get Floos means peace.
Comparing 1st and 3rd statements, we get Beiling means in
So, Hoomp is "we come".
Final answer:
"Hoomp" means "we come".
Explanation:
The question at hand involves deciphering the meaning of a word from messages in a strange language received from a distant planet.
By analyzing the given messages and their translations, we can determine the meaning of individual words through comparison and elimination.
Here's a step-by-step explanation:
"Floos Beiling Hoomp" means "We come in peace"."Moog Naline Floos" means "Peace, Love, and Freedom "."Beiling Boog Vladi" means "In the last hour".Comparing the first and second statements, we understand that "Floos" means peace. From the first and third statements, we deduce "Beiling" means in.
Therefore, given its unique presence in the first statement, "Hoomp" logically translates to "we come".
This question illustrates the process of deciphering language through pattern recognition and comparative analysis, which is a fundamental aspect of linguistics and communication studies.
Kate and Stella both worked at the coffee shop today. Kate's total cups of coffee made is represented by f(x); and Stella's total cups of coffee made is represented by g(x). Use the functions below to write a function that represents the total cups of coffee they made today.
f(x) = 6x − 8
g(x) = 3x + 1
Answer:
The function that represents the total cups of coffee they made today is [tex]h(x)=9x-7[/tex].
Step-by-step explanation:
The given function are
[tex]f(x)=6x-8[/tex]
[tex]g(x)=3x+1[/tex]
Where, Kate's total cups of coffee made is represented by f(x) and Stella's total cups of coffee made is represented by g(x).
The total cups of coffee they made today is represented by the function
[tex]h(x)=f(x)+g(x)[/tex]
Substitute the value of functions.
[tex]h(x)=6x-8+3x+1[/tex]
Combine like terms.
[tex]h(x)=(6x+3x)+(-8+1)[/tex]
[tex]h(x)=9x-7[/tex]
Therefore the function that represents the total cups of coffee they made today is h(x)=9x-7.
Answer:
its 9x-7
Step-by-step explanation:
Consider these scenarios.
1. An elephant weighs 1.5 × 104 units.
2. A mouse weighs 6.3 × 10-2 units.
3. A puppy weighs 1.2 × 102 units.
Determine the unit of measurement that best represents each scenario.
1.
2.
3.
Final answer:
The elephant is likely measured in pounds, as it fits the known weight range for adult elephants. The mouse is measured in a smaller unit such as grams due to its size. The puppy is best measured in pounds, a common unit for pet animals.
Explanation:
In mathematics, specifically in the context of measurement, it is important to choose the most appropriate unit of measurement depending on the size of the object or substance being measured. When we look at the example scenarios, we can determine the best units as follows:
An elephant weighs 1.5 × 10⁴ units. Given that the range of weight for a male African elephant is between 12,000 and 16,000 pounds, and considering that 1.5 × 104 is 15,000, which is within this range, it is most likely that the unit of measurement in this scenario is pounds.A mouse weighs 6.3 × 10⁻² units. Since a mouse is a very small creature, a larger unit like pounds would be impractical. Generally, rodents are weighed in grams, a smaller and manageable unit of measurement more appropriate for small animals.A puppy weighs 1.2 × 10² units. A puppy, larger than a mouse but much smaller than an elephant, would be best measured in pounds, as that is a common unit of weight for domestic animals like dogs.2 Geometry Questions thank you guys :):)
Answer:
[tex]\large\boxed{Q5.\ x=45\sqrt2}\\\boxed{Q6.\ x=8\sqrt2,\ y=4\sqrt6}[/tex]
Step-by-step explanation:
Q5.
x it's a diagonal of a square.
The formula of a length of diagonal of a square:
[tex]d=a\sqrt2[/tex]
a - side of a square
We have a = 45.
Substitute:
[tex]x=45\sqrt2[/tex]
Q6.
Look at the first picture.
In a triangle 45° - 45° - 90°, all sides are in ratio 1 : 1 : √2.
In a triangle 30° - 60° - 90°, all sidea are in ratio 1 : √3 : 2.
Look at the second picture.
from the triangle 45° - 45° - 90°
[tex]a\sqrt2=8[/tex] multiply both sides by √√2 (use √a · √a = a)
[tex]2a=8\sqrt2[/tex] divide both sides by 2
[tex]a=4\sqrt2[/tex]
from the triangle 30° - 60° - 90°
[tex]x=2a\to x=2(4\sqrt2)=8\sqrt2[/tex]
[tex]y=a\sqrt3\to y=(4\sqrt2)(\sqrt3)=4\sqrt6[/tex]
Answer:
6. [tex]\displaystyle 4\sqrt{6} = y \\ 4\sqrt{2} = x[/tex]
5. [tex]\displaystyle 45\sqrt{2} = x[/tex]
Step-by-step explanation:
30°-60°-90° Triangles
Hypotenuse → 2x
Short Leg → x
Long Leg → x√3
45°-45°-90° Triangles
Hypotenuse → x√2
Two identical legs → x
6. You solve the shorter triangle first:
[tex]\displaystyle a^2 + b^2 = c^2 \\ \\ \\ x^2 + x^2 = 8^2 \\ \\ \frac{2x^2}{2} = \frac{64}{2} → \sqrt{x^2} = \sqrt{32} \\ \\ 4\sqrt{2} = x[/tex]
Now that we know our x-value, we can solve the larger triangle:
[tex]\displaystyle 4\sqrt{6} = 4\sqrt{2}\sqrt{3} \\ \\ 4\sqrt{6} = y[/tex]
5. This exercise is EXTREMELY SIMPLE since two congruent isosceles right triangles form that square, so all you have to do, according to the rules for 45°-45°-90° triangles, is attach [tex]\displaystyle \sqrt{2}[/tex]to 45, giving you [tex]\displaystyle 45\sqrt{2}.[/tex]
I am joyous to assist you anytime.
Solve sin 0 + 1 = cos20 on the interval 0 ≤ 0 < 2pi. Show work please!
Answer:
[tex]\theta=\frac{\pi}{2},\frac{3\pi}{2}\frac{2\pi}{3}\frac{4\pi}{3}[/tex]
Step-by-step explanation:
You need 2 things in order to solve this equation: a trig identity sheet and a unit circle.
You will find when you look on your trig identity sheet that
[tex]cos(2\theta)=1-2sin^2(\theta)[/tex]
so we will make that replacement, getting everything in terms of sin:
[tex]sin(\theta)+1=1-2sin^2(\theta)[/tex]
Now we will get everything on one side of the equals sign, set it equal to 0, and solve it:
[tex]2sin^2(\theta)+sin(\theta)=0[/tex]
We can factor out the sin(theta), since it's common in both terms:
[tex]sin(\theta)(2sin(\theta)+1)=0[/tex]
Because of the Zero Product Property, either
[tex]sin(\theta)=0[/tex] or
[tex]2sin(\theta)+1=0[/tex]
Look at the unit circle and find which values of theta have a sin ratio of 0 in the interval from 0 to 2pi. They are:
[tex]\theta=\frac{\pi}{2},\frac{3\pi}{2}[/tex]
The next equation needs to first be solved for sin(theta):
[tex]2sin(\theta)+1=0[/tex] so
[tex]2sin(\theta)=-1[/tex] and
[tex]sin(\theta)=-\frac{1}{2}[/tex]
Go back to your unit circle and find the values of theta where the sin is -1/2 in the interval. They are:
[tex]\theta=\frac{2\pi}{3},\frac{4\pi}{3}[/tex]
Help plz system of inequalities
Answer:
c. 1 ≤ y ≤ 3; y ≥ -2x +2
Step-by-step explanation:
The shading between the solid lines y=1 and y=3 tells you that one of the inequalities is 1 ≤ y ≤ 3 (including the "or equal to" case). The shading above the solid diagonal line tells you that inequality will be y ≥ (something).
Choice C matches these observations.
A family is traveling in a car at a constant average speed during a road trip. The function d(t)=70t+620 models the distance d, in miles, the family is from their house t hours after starting to drive on the second day of the road trip.
A) At what average speed is the family's car traveling?
-Explain
B) What is the distance between the family's house and the point where they started driving on the second day
-Explain
Answer:
A. 70 miles per hour B. 620 miles from home
Step-by-step explanation:
This function is a linear equation, following the slope-intercept form of a line. This standard form is y = mx + b, where m is the slope and b is the y-intercept. The slope of a line is the rate at which the steepness of the line is changing. The y-intercept is where the function is "starting".
In our case, the number in the rate of change position is 70. It is being multiplied by t. If t = 1, that means that after 1 hour, we have gone 70 miles. If t = 2, that means after 2 hours we have gone 140 miles. If t = 3, that means that after 3 hours, we have gone 210 miles; etc. That number in the slope position represents the rate at which you are traveling PER HOUR; the slope.
The "starting" position of day 2 is found in the y-intercept. Replacing x with 0, meaning NO time has gone by at all, at the beginning of the second day, we are starting 620 miles from home.
The family's car is traveling at an average speed of 70 miles per hour. The distance between the family's house and the starting point on the second day is 620 miles.
The given function d(t) = 70t + 620 models the distance in miles from the family's house t hours after starting to drive on the second day of their road trip.
A) At what average speed is the family's car traveling?
The coefficient of t in the distance function, which is 70, represents the family's car average speed. Therefore, the car is traveling at an average speed of 70 miles per hour.
B) What is the distance between the family's house and the point where they started driving on the second day?
The constant term in the distance function, which is 620, signifies the distance in miles from the family's house at t = 0 or the starting point. Thus, the distance between the family's house and the starting point on the second day is 620 miles.
Can u guys PLEASE do this question 30 a
Answer:
1/2 units (I'm not sure what units the scale is using).
Step-by-step explanation:
We could setup a proportion. The trick to doing this is lining up corresponding parts.
The scale of a plan is 1 to 200.
We want to know the scale distance that represents the distance 100 m.
So we have:
1 to 200
x to 100
Setting up a proportion:
[tex]\frac{1}{x}=\frac{200}{100}[/tex]
Cross multiply:
[tex]100(1)=200(x)[/tex]
Divide both sides by 200:
[tex]\frac{100(1)}{200}=x[/tex]
[tex]\frac{100}{200}=x[/tex]
[tex]\frac{1}{2}=x[/tex]
[tex]x=\frac{1}{2}[/tex]
So a 1/2 units in length on the scale represents 100 m.
In how many ways can we put five identical fruits into three bowls? Note that the bowls may be empty.
The number of ways to distribute five identical fruits into three bowls is solved using the stars and bars technique in combinatorics, resulting in 21 different ways.
Explanation:The student is asking about the number of ways to distribute five identical fruits into three bowls, where bowls can be left empty.
This problem is a classic example of a combinatorial problem in mathematics, often approached using the stars and bars method.
The stars and bars technique is a way to solve problems related to distributing indistinguishable items into distinguishable groups.
To solve this, think of the five identical fruits as stars (*) and the separations between bowls as bars (|).
We need to place two bars to create three sections (bowls) among the five stars.
The question then becomes: In how many ways can we arrange five stars and two bars?
This is equivalent to choosing two places for bars out of seven possible positions (five stars plus two bars).
The number of ways we can choose two positions out of seven for the bars is given by the combination formula C(n, k) = n! / (k!(n - k)!), where n is the total number of items and k is the number of items to choose.
In this case, C(7, 2) = 7! / (2!5!) = 21 ways.
Therefore, there are 21 different ways to distribute five identical fruits into three bowls.
Sharla invests $275 in a simple interest bearing account for 16 years. The annual interest rate is 8%. Using the simple interest formula, how much interest will Sharla’s initial investment earn over the 16 year period?
$297
$319
$352
$627
Answer:
$352
Step-by-step explanation:
Simple interest (I) is calculated as
I = [tex]\frac{PRT}{100}[/tex]
where P is the principal ( investment), R is the rate of interest and T is time in years, thus
I = [tex]\frac{275(8)(16)}{100}[/tex] = $352
Answer:
$352
Step-by-step explanation:
Find the value for tan θ given the point (-3, 4) on the terminal side. Leave your answer in fraction form
Find the value for sec θ given the point (-3, 4) on the terminal side. Leave your answer in fraction form.
Answer:
Step-by-step explanation:
The point (-3, 4) is in QII. If we plot this point and drop an altitude then connect the point to the origin, we have a right triangle with side opposite measuring 4 units and side adjacent measuring |-3|. The tangent of the reference angle is the ratio side opposite/side adjacent, so
[tex]tan\theta=-\frac{4}{3}[/tex]
Since secant is the reciprocal of cosine, let's find the cosine of the reference angle and then flip it upside down. The cosine of the angle is the side adjacent (got it) over the hypotenuse (don't have it). We can find the hypotenuse using Pythagorean's Theorem:
[tex]c^2=-3^2+4^2[/tex] s0
[tex]c^2=25[/tex] and
c = 5
The cosine of the angle theta is
[tex]cos\theta=-\frac{3}{5}[/tex]; therefore,
[tex]sec\theta=-\frac{5}{3}[/tex]
In the figure below, angle B and arc AC are congruent.
Answer:
False
Step-by-step explanation:
Correct me if I'm wrong but this suppose to be a true or false question.
The measure of angle of B is going to be half the arc measure of AC so they don't have the same measurement. If they don't have the same measurement, then they can't be congruent.
let f(x)=3x+5 and g(x)=x^2 find f(x) + g(x)
A) x^2 +3x+5
B) x^3+5
C)3x^2+5
D)3x^3 +5x^2
Answer:
[tex]\large\boxed{A).\,x^2+3x+5}[/tex]
Step-by-step explanation:
In this question, we're trying to figure out what f(x) + g(x) equals to.
We are going to be plugging in the equations they gave to us and solve.
Equations we're going to use:
f(x)=3x+5g(x)=x^2Now, lets get to solving.
We would plug the equations in the appropriate spot. In other words, we're pretty much plugging them in to the right variable.
Your expression should look like this:
[tex]3x+5+x^2[/tex]
Now we solve.
[tex]3x+5+x^2\\\\x^2\,\text{can't combine with 3x because of the exponent, so we will eave it as is}\\\\\text{There is nothing else we can combine, so it would stay as:}\\\\x^2+3x+5[/tex]
You should end up with x²+3x+5.
This means that your final answer would be A) x^2 +3x+5
I hope this helped you out.Good luck on your academics.Have a fantastic day!Laura has pledges of $5 for each mile she walks in the Juvenile Diabetes Walkathon fundraiser.
-Use the table below to record the miles walked and the money earned for miles 0 through 6
- Write a rule relating miles walked to money collected
Answer:
r = 5m
Step-by-step explanation:
miles walked money earned
0 0
1 1*5 5
2 2 * 5 10
3 3 * 5 15
4 4 * 5 20
5 5 * 5 25
6 6 * 5 30
We multiply 5 by each mile walked to determine how much money was raised
r = 5m where r is the money raised and m is the miles walked
Sebastian has just graduated after four years of university. He took out an unsubsidized Stafford loan worth $8,180 to help pay for his tuition. The loan has a duration of ten years. If the loan has an interest rate of 5.3%, compounded monthly, how much interest capitalization has occurred by the time he graduated? Round all dollar values to the nearest cent.
a.
$721.93
b.
$842.25
c.
$1,926.97
d.
$1,734.24
Answer:
The answer would be C. 1,926.97 :)
Answer:
Option C
Step-by-step explanation:
Principal,P = $8180
Time = 10 years
Rate,R = 5.3% compounded monthly
Formula: [tex]P(1+\frac{R}{n} )^{nt}[/tex], where n is the number of times interest is compounded that is monthly, quarterly, yearly.
Now, we have to calculate the interest by the time Sebastian has graduated that is the time now is 4 years, t = 4 and n = 12, because the interest is compounded monthly.
Puttin all the values in the formula
Amount = [tex]8180(1+ \frac{5.3}{1200} )^{48}[/tex] = 10106.9707$
Also, Interset = Amount - Principal
Interest = 10106.9707$ - 8180$ = 1926.97074$
Option C
Please help me with this problem, lots of points!
Answer:
A f(x) = 1/3x^3
Step-by-step explanation:
g(x) = x^3
We want f(x) to be vertically compressed
f(x) = ag(x )
a is vertical stretch/compression
|a| > 1 stretches
|a| < 1 compresses
so we need to multiply by a number less than between 0 and 1
If a is negative it is a reflection, which was not asked for
f(x) = 1/3 x^3
Answer:
A f(x) = 1/3x^3
Step-by-step explanation:
g(x) = x^3
We want f(x) to be vertically compressed
f(x) = ag(x )
a is vertical stretch/compression
|a| > 1 stretches
|a| < 1 compresses
so we need to multiply by a number less than between 0 and 1
If a is negative it is a reflection, which was not asked for
f(x) = 1/3 x^3
Benito spent $1822 to operate his car last year. Some of these expenses are listed below. Benito's only other expense was for gasoline. If he drove 7340 miles, what was the average cost of the gasoline per mile?
Operating Expenses
Insurance $809
Registration $170
Maintenance $57
A. 61.06 cents
B. 10.71 cents
C. 1.27 cents
D. not enough information
Answer:
B. 10.71 cents
Step-by-step explanation:
Benito's other expenses total ...
$809 +170 +57 = $1036
so his gas expense is ...
$1822 -1036 = $786
Then the per-mile cost is ...
$786/(7340 mi) ≈ $0.10708/mi
Benito's gas cost averaged 10.71¢ per mile.
Answer is B) 10.71 cents.
You just add Benito's operating expenses togther, like so:
809 + 170 + 57 = $1036 so far.
Next, subtract the total amount he spent this last year (1822), with how much he spent so far (1036).
1822 - 1036 = $786
Finally, divide the final total with the 7340 miles he plans to drive.
786 ÷ 7340 = 0.10708
Answer is B) 10.71 cents.
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Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 45 degrees at midnight and the high and low temperature during the day are 50 and 40 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.
Answer:
so equation for the temperature in terms of t is D = 5 sin [tex]\pi[/tex] / 12 (t) + 45
Step-by-step explanation:
Given data
temperature = 45 degrees
high temperature = 50 degrees
low temperature = 40 degrees
to find out
an equation for the temperature in terms of t
solution
first we find the amplitude i.e.
Amplitude (A) = ( high temperature - low temperature ) / 2
Amplitude (A) = (50 - 40) / 2
Amplitude (A) = 5
here we know in a day 24 hours so
2[tex]\pi[/tex] /K = 24
K = [tex]\pi[/tex] / 12
so we have temperature equation is
temperature D = amplitude sinK (t) + avg temperature midnight
D = 5 sin [tex]\pi[/tex] / 12 (t) + 45
so equation for the temperature in terms of t is D = 5 sin [tex]\pi[/tex] / 12 (t) + 45
The temperature over a day can be modeled as a sinusoidal function. The equation for the temperature, D, in terms of t is: D = 5cos((pi/12)t) + 45.
Explanation:The temperature over a day can be modeled as a sinusoidal (sine or cosine) function. To find an equation for the temperature, we can use the cosine function because it starts at its maximum value at t = 0, which corresponds to midnight. The equation for the temperature, D, in terms of t is:
D = 5cos((pi/12)t) + 45
Here, t represents the number of hours since midnight, and D represents the temperature in degrees. The amplitude of the sinusoidal function is 5, which represents the difference between the high and low temperatures. The cosine function is scaled and shifted to match the given data: it is multiplied by 5 to match the amplitude, and 45 is added to shift the function vertically so that it starts at 45 degrees at t = 0.
Learn more about Temperature modeling here:https://brainly.com/question/33485356
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MAJORR HELP!!!!
In the graph, what are the x- and y-coordinates of the center?
Answer:
D: (-1,2)
Step-by-step explanation:
The X coordinate is between 2 and -4. There is a difference of 6, so you should do -4 + (6/2) = -1.
so the X coordinate is -1
The y coordinate is between 0 and 4. There is a difference of 4, so you should do 0 + (4/2) = 2.
so the y coordinate is 2
This results in the centre being (-1,2)
The coordinates of the center of the equation is (- 1, 2).
We have a ellipse in the figure.
We have to find out the coordinates of the center of ellipse (x, y).
What is an general equation of Ellipse?The general equation of an ellipse in the rectangular coordinate system is -
[tex]\frac{x^{2} }{a^{2} } +\frac{y^{2} }{b^{2} } = 1[/tex]
In the figure given to us, the center of the ellipse at the point of intersection of the lines of equation -
x + 1 = 0
and
y - 2 = 0
The coordinates of the center -
x = - 1 and y = 2.
Hence, the coordinates of the center of the ellipse is (- 1, 2).
To solve more questions on ellipse, visit the link below -
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Tony is 4 years younger than his brother josh and two years older than his sister Cindy. Tony also has a twin brother, Evan. All the ages totaled together of the children is 66. How old is tony?
Let Tony's age = x
He is 4 years younger than his brother Josh, so Josh's age would be x + 4
He is 2 years older than his sister, so her age would be x - 2
He has a twin, which would be the same age, so the twins age is also x
They all add together to equal 66, so you get:
x + x + x+4 + x-2 = 66
Simplify:
4x +2 = 66
Subtract 2 from both sides:
4x = 64
Divide both sides by 4:
x = 64/4 = 16
Tony is 16 years old.
Final answer:
By setting up an equation with Tony's age as T, and considering the ages of his siblings, we find that Tony is 16 years old.
Explanation:
Tony is 4 years younger than his brother Josh and two years older than his sister Cindy. Additionally, Tony has a twin brother, Evan. If we add the ages of Tony, Josh, Evan, and Cindy, the total is 66 years. To solve for Tony's age, we need to set up an equation. Let's assume Tony's age is T years.
Since Tony is 4 years younger than Josh, Josh is T + 4 years old. As Tony has a twin brother, Evan, Evan is also T years old. Cindy, being two years younger than Tony, is T - 2 years old. So, the sum of all their ages is:
T + (T + 4) + T + (T - 2) = 66
Combining like terms, we get:
4T + 2 = 66
Subtracting 2 from both sides gives us:
4T = 64
Dividing both sides by 4 results in:
T = 16
Therefore, Tony is 16 years old.