Answer:
a one-unit increase
Step-by-step explanation:
In a simple regression model, the relationship between x and y can be represented by the equation y = ax+b, where
a is the slopeb is the y-interceptThe slope term is the change in the mean value of y associated with a one-unit increase in x.
A deck is shuffled and two cards are dealt. (a) Find the chance that the second card is a heart given the first card is a heart. (b) Find the chance that the first card is a heart and the second card is a heart.
Answer: a) 12/51
b) 3/51
Step-by-step explanation:
we are assuming the cards are dealt without replacement
a) given that the first card is heart, we are left with 12 hearts and 51 cards in total
Therefore, the probability that the second card is heart is:
P2 = 12/51
b) probability that the two cards are hearts is given by:
P = 13/52 * 12/51
P = 3/51
(a) The chance that the second card is a heart given the first card is a heart is [tex]\(\frac{12}{51}\)[/tex].
(b) The chance that the first card is a heart and the second card is a heart is [tex]\(\frac{1}{17}\)[/tex].
(a) To find the probability that the second card is a heart given that the first card is a heart, we use conditional probability. There are 13 hearts in a standard deck of 52 cards. Once the first heart is drawn, there are 12 hearts left and the total number of cards left is 51. The probability of drawing a heart as the second card, given that the first card is a heart, is the number of remaining hearts divided by the total number of remaining cards.
[tex]\[ P(\text{second card is a heart | first card is a heart}) = \frac{12}{51} \][/tex]
This simplifies to:
[tex]\[ P(\text{second card is a heart | first card is a heart}) = \frac{4}{17} \][/tex]
(b) To find the probability that both the first and second cards are hearts, we multiply the probability of drawing a heart first by the probability of drawing a heart second given that the first card is a heart. The probability of drawing a heart first is [tex]\(\frac{13}{52}\)[/tex], which simplifies to [tex]\(\frac{1}{4}\)[/tex]. We already calculated the probability of drawing a heart second given a heart first as [tex]\(\frac{12}{51}\) or \(\frac{4}{17}\)[/tex].
[tex]\[ P(\text{first card is a heart and second card is a heart}) = P(\text{first card is a heart}) \times P(\text{second card is a heart | first card is a heart}) \][/tex]
[tex]\[ P(\text{first card is a heart and second card is a heart}) = \frac{1}{4} \times \frac{4}{17} \][/tex]
[tex]\[ P(\text{first card is a heart and second card is a heart}) = \frac{1}{17} \][/tex]
Thus, the probability that the first card is a heart and the second card is a heart is [tex]\(\frac{1}{17}\)[/tex].
In Andrew’s Furniture Shop, he assembles both bookcases and TV stands. Each type of furniture takes him about the same time to assemble. He figures he has time to make at most 18 pieces of furniture by this Saturday. The materials for each bookcase cost him $20.00 and the materials for each TV stand cost him $40.00. He has $600.00 to spend on materials. Andrew makes a profit of $60.00 on each bookcase and a profit of $100.00 for each TV stand. Find how many of each piece of furniture Andrew should make so that he maximizes his profit.
Answer:
6 bookcases12 TV standsStep-by-step explanation:
Given Andrew has $600 for materials and can make 18 pieces of furniture, you want to know the number of each kind that maximizes profit if each bookcase costs $20 and gives $60 profit, while each TV stand costs $40 and gives $100 profit.
SetupIf x and y represent the numbers of bookcases and TV stands Andrew builds, respectively, then he wants to ...
maximize 60x +100y
subject to ...
x + y ≤ 1820x +40y ≤ 600SolutionThe attached graph shows the solution space for these constraints. The profit is maximized at the vertex of the space where the profit function line is farthest from the origin. Andrew maximizes his profit by building ...
6 bookcases12 TV standsAndrew needs to solve a linear programming problem to find how many bookcases and TV stands he should manufacture for optimal profit. This is done by setting up and solving inequalities representing Andrew's time and material cost constraints, graphing the feasible region, and finding the point(s) in this region that yield the highest profit.
Explanation:This question deals with the topics of linear programming and profit maximisation. Here, Andrew has to decide how much of each type of furniture, bookcases or TV stands, he should produce to maximise profit while considering time and material cost constraints.
From the given conditions, we get two inequalities. The first related to time says that the total number of bookcases and TV stands is less than or equal to 18: let bookcases be x, TV stands be y, thus we have x + y <= 18. The second involving the cost of material says that the total cost spent on materials for both products does not exceed $600: thus, we also have 20x + 40y <= 600.
You can graph these inequalities on the x-y plane to get a visual representation of the possibilities.
Finally, to find the optimal solution (i.e., the highest profit), you calculate the profit function P = 60x + 100y for each point in the feasible region and select the point that provides the highest profit.
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A monopolist faces the following demand curve, marginal revenue curve, total cost curve and marginal cost curve for its product: Q = 200 - 2P MR = 100 - Q TC = 5Q MC = 5 Refer to Scenario 10.2. What is the profit maximizing level of output?
Answer:
Q=95, P= 52.5
Step-by-step explanation:
The profit maximizing level of output in monopolies is reached when the marginal cost is equal to the marginal revenue. This is also the profit maximizing rule in perfect competition, the difference between both is that is perfect competition the marginal revenue is equal to the price while in monopolies, the demand curve is often above the marginal revenue curve, then the actual price (defined by the demand curve) is often higher than the marginal revenue price.
For this problem the profit maximizing level of output is:
MC=MR
5=100-Q
Q=95
Because monopolies decide the selling price based on the demand curve, you should replace this quantity in the demand curve equation:
95=200-2P
95-200/2=-P
P= 52.5
When output is set so that marginal revenue and marginal cost are equal in this economics dilemma, the monopolist will make the most money. That would be at a manufacturing level of 95 units in this instance.
Explanation:When establishing output, a monopolist in economics attempts to maximize profit by ensuring that Marginal Cost (MC) and Marginal Revenue (MR) are equal. We must assign MR to MC in this case given that MR = 100 - Q and MC = 5. Therefore, 100 - Q = 5, giving Q=95. In order to maximize its profit, the monopolist should create 95 units of the good.
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Students were surveyed about their favorite colors 1/4 of the students preferred read 1/8 of the students preferred blue and 3/5 of the remaining students were for green if 15 students prefer green how many students were surveyed
Answer: the number of students that were surveyed is 40
Step-by-step explanation:
Let x = total number of students that were surveyed about their favorite colors
1/4 of the students preferred red.
This means that the number of students that preferred red is 1/4 × x = x/4
1/8 of the students preferred blue.
This means that the number of students that preferred blue is 1/8 × x = x/8
The remaining number of students will be the total number of students - the sum of the number of students that preferred red and the number of students that preferred blue. It becomes
x - (x/4 + x/8) = x - 3x/8 = 5x/8
3/5 of the remaining students were for green. This means that the number of students that preferred green is 3/5 × 5x/8 = 3x/8
if 15 students prefer green, then
3x/8 = 15
3x = 120
x = 120/3 = 40 students
Omar picks 7 pounds of cherries . He divides them into containers that holds one whole 1/4 pounds of blue berries each. How many containers can he fill?
Answer:
Omar can fill 28 containers
Step-by-step explanation:
Omar have 7 pounds of cherries, each pound need 4 containers because each one only hold of [tex]\frac{1}{4}\\[/tex] pound, now we multiplicate 7 pounds with 4 container for each one and we get 28 containers.
[tex]Number containers = \frac{7 pounds}{\frac{1}{4}pounds } = 28[/tex]
The number of wiring packages that can be assembled by a company's employees has a normal distribution, with a mean equal to 19.8 per hour and a standard deviation of 1.2 per hour.
(a) What are the mean and standard deviation of the number X of packages produced per worker in an 8-hour day? (Round your standard deviation to three decimal places.)
a. mean
b. standard deviation
(b) Do you expect the probability distribution for X to be mound-shaped and approximately normal? Explain.
Choose
A. Yes, since the standard deviation is less than 3, the sampling distribution of the sum will be approximately normal.
B. Yes, since the original population is normal, the sampling distribution of the sum will also be approximately normal.
C. No, since the original population is not normal, the sampling distribution of the sum will not be approximately normal.
D. No, since the original population is normal, the sampling distribution of the sum cannot be normal.
E. No, since the standard deviation is more than 3, the sampling distribution of the sum will not be approximately normal.
(c) What is the probability that a worker will produce at least 160 packages per 8-hour day? (Round your answer to four decimal places.)
Answer:
a) mean= 158.4 , standard deviation = 3.394
b) Best option : B. Yes, since the original population is normal, the sampling distribution of the sum will also be approximately normal.
c) P(X>160) = P(Z>0.471) = 1-P(Z<0.471) = 0.3188
Step-by-step explanation:
1) Notation
n = sample size = 8
[tex] \mu [/tex] = population mean = 19.8
[tex] \sigma [/tex] = population standard deviation = 1.2
2) Definition of the variable of interest
Part a
The variable that we are interested is [tex] \sum x_i [/tex] and the mean and the deviation for this variable are given by :
E([tex] \sum x_i [/tex]) = [tex] \sum E(x_i) [/tex] = n [tex] \mu [/tex] = 8x19.8 = 158.4
Var([tex] \sum x_i [/tex]) = [tex] \sum Var(x_i) [/tex] = n [tex] \sigma^2 [/tex]
Sd([tex] \sum x_i [/tex]) = [tex] \sqrt{n \sigma^2} [/tex] = [tex] \sqrt(8) [/tex] x 1.2 = 3.394
Part b
For this case the populations are normal, then the distribution for the sample ([tex] \sum x_i [/tex]) is normal too.
Based on this the distribution for the variable X would be normal, so the best option should be:
B. Yes, since the original population is normal, the sampling distribution of the sum will also be approximately normal.
Part c
From part a we know that the mean = 158.4 and the deviation = 3.394
The z score is defined as
Z = (X -mean)/ deviation = (160-158.4)/ 3.394 = 0.471
Then we can find the probability P(X>160) = P(Z>0.471) = 1-P(Z<0.471) = 0.3188
The width of a small playground 3 meters less than the length of the playground. If the area of the playground 54 square meters. What are the dimensions of the playground.
Answer: Length = 9 meters
Width = 6 meters
Step-by-step explanation:
The diagram of the playground is shown in the attached photo
The width of a small playground 3 meters less than the length of the playground.
Length of playground = L meters
Width of playground = (L-3) meters
The area of the playground is 54 square meters.
Area = L × W
54 = L(L - 3)
54 = L^2 - 3L
L^2 - 3L - 54 = 0
L^2 + 6L - 9L- 54 = 0
L(L+6) - 9(L+6) = 0
(L-9)(L+6) = 0
L -9 = 0. or L+6 = 0
L = 9 or L = -6
L cannot be negative so, the length of the playground is 9 meters
The width of the playground is
L-3 = 9-3 = 6 meters
A pilot has just started on the glide path for landing at an airport with a runway of length 9000 feet. The angles of depression from the plane to the ends of the runway are 17.5 degrees and 18.5 degrees.
(a) Draw a diagram that visually represents the problem
(b) Find the air distance the plane must travel until touching down on the near end of the runway.
(c) Find the ground distance the plane must travel until touching down
(d) Find the altitude of the plane when the pilot begins the descent
Answer:
a. See attachments
B. 163630.2ft
C. 147056.70ft
D. 49205.4ft
Step-by-step explanation:
Check attachments for details
Angle a.. Angle of dep to beginning of runway
Angle b... Angle of del to end of runway
AD... Altitude of plane
DB..... Ground distance before touch down
AC... Air distance travelled until touch down on the near end of runway
The sum of the first ten terms of a particular arithmetic sequence is four times the sum of the first five terms of the sequence. What is the ratio of the first term to the second term? Express your answer as a common fraction.
In the given arithmetic sequence, the ratio of the first term to the second term is 1:2.
Explanation:The problem is based on the properties of the arithmetic sequence. To solve it, we use the formula for the sum of an arithmetic sequence: S = n/2*(a + l), where n is the number of terms, a is the first term, and l is the last term.
From the question, we know that 4 times the sum of the first five terms equals the sum of first ten terms. Therefore, 4 * (5/2 * (a + a + 4d)) = 10/2 * (a + a + 9d), where d is the common difference. Simplifying, we find that the ratio of the first term a to the second term (a + d) is 1:2.
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The school system is trying to help their students be more successful. They are going to hire tutors to help. They must have more teachers than tutors. The maximum number of teachers and tutors to be hired is 50. How many tutors and how many teachers can they hire? Let x = number of teachers hired. Let y = number of tutors hired. a. Write an inequality that represents the statement that the number of teachers hired must exceed the number of tutors hired. (2 points) b. Write an inequality that represents the statement that the maximum number of teachers and tutors is 50. (2 points) c. Choose a point that satisfies the situation, and explain why you chose that number of tutors and teachers. (1 point)
Answer:
a. [tex]x-y > 0[/tex]
b. [tex]x+y \leq 50[/tex]
c. x=26; y=24 or (24,26)
Step-by-step explanation:
Let x = number of teachers hired.
Let y = number of tutors hired.
Now solving for part a we get
a. Write an inequality that represents the statement that the number of teachers hired must exceed the number of tutors hired.
[tex]x-y > 0[/tex]
solving for part b we get;
b. Write an inequality that represents the statement that the maximum number of teachers and tutors is 50.
[tex]x+y \leq 50[/tex]
solving for part c we get;
c. Choose a point that satisfies the situation, and explain why you chose that number of tutors and teachers.
Now we know that [tex]x-y > 0[/tex] also [tex]x+y \leq 50[/tex]
x=26; y=24
(24,26)
Explanation: To make number of teacher more than number of tutors this is the maximum value we can achieve for the requirement.
a. The inequality x > y represents the statement that the number of teachers hired must exceed the number of tutors hired. b. The inequality x + y ≤ 50 represents the statement that the maximum number of teachers and tutors is 50. c. The point (25, 10) satisfies both inequalities and represents hiring 25 teachers and 10 tutors.
Explanation:a. To represent the statement that the number of teachers hired must exceed the number of tutors hired, you can write the inequality x > y. This means that the number of teachers, represented by x, must be greater than the number of tutors, represented by y.
b. To represent the statement that the maximum number of teachers and tutors is 50, you can write the inequality x + y ≤ 50. This means that the total number of teachers and tutors hired, represented by x + y, cannot exceed 50.
c. Let's choose the point (25, 10) which represents hiring 25 teachers and 10 tutors. This point satisfies both inequalities: 25 > 10 and 25 + 10 ≤ 50.
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Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?
(A) x/(x+y)
(B) y/(x+y)
(C) xy/(x+y)
(D) xy/(x-y)
(E) xy/(y-x)
Answer:
(E) xy/(y-x)
Step-by-step explanation:
The sum of the individual rates is the total rate. Each machine's rate is nails per hour is the inverse of its rate in hours per nail.
total rate = (800 nails)/(x hours) = 800/x nails/hour
A rate = (800 nails)/(y hours) = 800/y nails/hour
We want to find the time "b" such that ...
B rate = (800 nails)/(b hours) = 800/b nails/hour
__
As we said, the total rate is the sum of the individual rates:
800/x = 800/y + 800/b
Multiplying by xyb/800, we get
yb = xb + xy
Solving for b, we have ...
yb -xb = xy
b(y -x) = xy
b = xy/(y-x) . . . . . matches choice E
It takes Machine B xy/(y-x) hours to produce 800 nails.
1. In triangle XYZ, A is the midpoint of XY, B is the midpoint of YZ, and C is the midpoint of XZ. Also, AY = 7, BZ = 8, and XZ = 18. What is the perimeter of triangle ABC? (SHOW WORK)
2. What is y? (SHOW WORK) 2nd picture is the triangle.
Answer:
Part 1) The perimeter of triangle ABC is 24 units
Part 2) [tex]y=97\°[/tex]
Step-by-step explanation:
Part 1) we know that
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side
see the attached figure to better understand the problem
so
Applying the midpoint theorem
step 1
Find the value of BC
[tex]BC=\frac{1}{2}XY[/tex]
[tex]XY=2AY[/tex] ---> because A is the midpoint
substitute the given value of AY
[tex]XY=2(7)=14\ units[/tex]
[tex]BC=\frac{1}{2}(14)=7\ units[/tex]
step 2
Find the value of AC
[tex]AC=\frac{1}{2}YZ[/tex]
[tex]YZ=2BZ[/tex] ---> because B is the midpoint
substitute the given value of BZ
[tex]YZ=2(8)=16\ units[/tex]
[tex]AC=\frac{1}{2}(16)=8\ units[/tex]
step 3
Find the value of AB
[tex]AB=\frac{1}{2}XZ[/tex]
substitute the given value of XZ
[tex]AB=\frac{1}{2}(18)=9\ units[/tex]
step 4
Find the perimeter of triangle ABC
[tex]P=AB+BC+AC[/tex]
substitute
[tex]P=9+7+8=24\ units[/tex]
Part 2) Find the measure of angle y
step 1
Find the measure of angle z
we know that
The sum of the interior angles in a triangle must be equal to 180 degrees
so
[tex]55\°+42\°+z=180\°[/tex]
solve for z
[tex]97\°+z=180\°[/tex]
[tex]z=180\°-97\°[/tex]
[tex]z=83\°[/tex]
step 2
Find the measure of angle y
we know that
[tex]z+y=180\°[/tex] ----> by supplementary angles (form a linear pair)
we have
[tex]z=83\°[/tex]
substitute
[tex]83\°+y=180\°[/tex]
solve for y
[tex]y=180\°-83\°[/tex]
[tex]y=97\°[/tex]
Ella can complete the cheep cheep beatch course in 1 minute and 25 seconds. If the course is 3 laps long. How long would it take her to race 2 laps? ( proportion)
Answer: it will take her 56.67 seconds or 0.945 minutes to race 2 laps
Step-by-step explanation:
It takes Ella 1 minute and 25 seconds to complete the cheep beach course.
We can express this time in seconds or minutes. Expressing it in seconds,
1 minute = 60 seconds
Therefore,
1 minute and 25 seconds = 60 +25 = 85 seconds
If the course is 3 laps long, that means she completed 3 laps in 85 seconds. The time it will take her to race 2 laps would be
(2 × 85)/3
= 56.67 seconds
Converting to minutes, it will be
56.7/60 = 0.945 minutes
Help asap please!!!!
A prism is made of cubes which measures 1/4 of an inch on one side. What is the volume?
A. 40 cubic in
B. 11/4 cubic in
C. 5/8 cubic in
D. 10 cubic in
Answer:
b
Step-by-step explanation:
The area of a triangle is 72 cm2. The height of the triangle is 8 cm. What is the measure of the base of the triangle? A. 18 cm B. 36 cm C. 27 cm D. 32 cm
Answer:
A. 18x8 is 144 then divided by 2 is 72
Answer:
Step-by-step explanation:
Simplify 15x^2-18/6
A. 15x+3
B. X+3
C. 5x^2-18/2
D. 5x^2-6/2
Answer:
D. 5x^2-6/2
Step-by-step explanation:
5x^2-18/6 , 18/6 is 6/2
5x^2-6/2 ,
After simplification, the equation obtained is 5x² - 6/2. Hence, option D is correct.
What is Simplification?A phrase must be simplified and made shorter using a variety of methods. The steps required to decrease something are carried out according to a specified order known as BOD MAS.
Make a fraction simpler by reducing it to the base form. If both the numerator and denominator of a fraction only include the integer, the fraction is considered to be in its basic form.
As per the given information in the question,
The given equation in the question is,
15x² - 18/6
Divide the equation by 3,
= 5x² - 6/2.
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determine the intervals on which the function is increasing, decreasing, and constant
Answer:
increasing: (-1, ∞)decreasing: (-∞, -1)constant: nowhereStep-by-step explanation:
A function is increasing when it is rising to the right. Here, that is everywhere right of x=-1.
A function is decreasing when it is falling to the right. Here, that is everywhere left of x=-1.
A function is constant when its graph is horizontal. There are no places on this graph like that.
Mrs. Martin directs two courses one chorus has 28 students the other chorus has 36 students for rehearsals she wants to divide each chorus into the largest possible equal groups with no students left over how many students will be in each group
Answer:4 students will be in each group.
Step-by-step explanation:
28 and 36. The 2 numbers are divisible by 4 without remainder
Answer:
4 students will be in each group.
Step-by-step explanation:
28 and 36. The 2 numbers are divisible by 4 without remainder
brainilest pl.z? i've never got brainilest.
Ian is a phycologist interested in determining the proportion of algae samples from a local rivulet that belonged to a particular phylum. A random sample of 50 alga were obtained and each was categorized as either being cyanobacteria or not. It was found that 38 were, in fact, cyanobacteria.Without relying on any previous knowledge Ian wanted to estimate the proportion that were cyanobacteria with a margin of error of at most 0.01 in a 99% confidence interval. How large a sample size would be required?
To estimate the proportion of cyanobacteria in algae samples with a 99% confidence level and a margin of error ≤ 0.01, Ian would need a sample size of at least 12111.
To determine the required sample size (n) for estimating the proportion of cyanobacteria in the algae samples with a margin of error of at most (0.01) and a (99%) confidence level, you can use the formula for the margin of error in estimating a population proportion:
[tex]\[ E = Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \][/tex]
Given:
- Margin of error (E) = 0.01
- Confidence level = (99%), which corresponds to a Z-score of approximately 2.576
- Estimated proportion [tex](\(\hat{p}\)) = \(\frac{38}{50} = 0.76\)[/tex] (from the given sample)
Now, the formula can be rearranged to solve for (n):
[tex]\[ n = \frac{\hat{p}(1-\hat{p})}{\left(\frac{E}{Z}\right)^2} \][/tex]
Substitute the given values:
[tex]\[ n = \frac{0.76 \times (1 - 0.76)}{\left(\frac{0.01}{2.576}\right)^2} \][/tex]
Now calculate:
[tex]\[ n \approx \frac{0.76 \times 0.24}{\left(\frac{0.01}{2.576}\right)^2} \][/tex]
n ≈ 12111
Therefore, Ian would need a sample size of at least 12111 algae samples to estimate the proportion of cyanobacteria with a margin of error of at most (0.01) in a (99%) confidence interval.
Ian would require a sample size of approximately 12,102 algae to estimate the proportion of cyanobacteria with a margin of error of at most 0.01 in a 99% confidence interval.
To determine the required sample size for estimating the proportion of algae samples that are cyanobacteria with a margin of error of at most 0.01 and a 99% confidence level, we can use the formula for the confidence interval for a proportion:
[tex]\[ n = \left(\frac{z_{\alpha/2} \cdot \sqrt{p(1-p)}}{E}\right)^2 \][/tex]
where:
- n is the sample size,
- [tex]\( z_{\alpha/2} \)[/tex] is the z-score corresponding to the desired confidence level (for 99%, [tex]\( z_{\alpha/2} = 2.576 \),[/tex]
- p is the estimated proportion of the population that has the characteristic of interest (in this case, being cyanobacteria),
- E is the margin of error.
The estimated proportion p can be taken from the initial sample of 50 algae, where 38 were cyanobacteria. Thus, pis estimated as:
[tex]\[ p = \frac{\text{Number of cyanobacteria}}{\text{Total sample size}} = \frac{38}{50} = 0.76 \][/tex]
Now, we plug in the values into the formula:
[tex]\[ n = \left(\frac{2.576 \cdot \sqrt{0.76(1-0.76)}}{0.01}\right)^2 \][/tex]
[tex]\[ n = \left(\frac{2.576 \cdot \sqrt{0.76 \cdot 0.24}}{0.01}\right)^2 \][/tex]
[tex]\[ n = \left(\frac{2.576 \cdot \sqrt{0.1824}}{0.01}\right)^2 \][/tex]
[tex]\[ n = \left(\frac{2.576 \cdot 0.427}{0.01}\right)^2 \][/tex]
[tex]\[ n = \left(\frac{1.101}{0.01}\right)^2 \][/tex]
[tex]\[ n = (110.1)^2 \][/tex]
[tex]\[ n \ = 12101.01 \][/tex]
Since we cannot have a fraction of an algae sample, we round up to the nearest whole number:
[tex]\[ n \ = 12102 \][/tex]
What value of x is in the solution set of 3(x-4) > 5x + 2?
Answer:
x < -7
Step-by-step explanation:
Eliminate parentheses:
3x -12 > 5x +2
-14 > 2x . . . . . . . . . add -2-3x
-7 > x . . . . . . . . . . . divide by 2
Any value of x less than -7 is in the solution set.
Please help !!! I’ll mark brainliest
A) State the chain rule for integration
Ans. The chain rule for integration is also known as " Integration by substitution "
Integration by substitution is taken in order to make integration solve easily in few steps.
For, [tex]I = \int\limits (x+2)^{2} \,dx[/tex]
Instead of expanding term [tex](x+2)^{2}[/tex]
With substitution of [tex]u = (x+2) [/tex] and [tex]du= 1 dx [/tex]
We simplified the integration as
[tex]I = \int\limits (u)^{2} \, du[/tex]
[tex]I = \frac{(u)^{3}}{3}+C[/tex]
By replacaing value of u=x+2
[tex]I = \frac{(x+2)^{3}}{3}+C[/tex]
B) State the rule of differentiation for the sine function.
Ans. We know that [tex]\frac{d}{dx}Sinx dx = Cosx [/tex]
C) Find the indefinite integral using substitution.
Ans.
Given, [tex]I = \int\limits {\frac{Cos14x}{Sin14x} } \, dx[/tex]
Take y = Sin14x
Differentiating both side
[tex]dy=14Cos14x dx [/tex]
[tex]\frac{dy}{14} = Cos14x\, dx[/tex]
Substituting values in integration,
[tex]I = \int\limits {\frac{Cos14x}{Sin14x} } \, dx[/tex]
[tex]I = \int\limits {\frac{1}{y} } \,\frac{dy}{14} [/tex]
[tex]I = \frac{1}{14}\int\limits {\frac{1}{y} } \,dy [/tex]
[tex]I = \frac{1}{14} lny + C [/tex]
Replacing values in the integration
[tex]I = \frac{1}{14} ln(14Sin14x) + C [/tex]
D)Check your work by taking a derivative of your answer from part C.
Ans.
Answer for Part C is [tex]I = \frac{1}{14} ln(14Sin14x) + C [/tex]
Differentiating the answer
we get,
[tex]=\frac{1}{14}\frac{d}{dx}[ ln(Sin14x) + C]\\=\frac{1}{14}\frac{1}{Sin14x} \frac{d}{dx}(Sin14x)+ \frac{d}{dx}C\\=\frac{1}{14}\frac{1}{Cos14x}(14Cos14x)\\=\frac{Cos14x}{Sin14x} \\ =I[/tex]
If 8 ounces of canned pumpkin has 82 calories ,how many calories are in one ounce? Use your answer to find how many calories are in 6 ounces of pumkin.
Number of calories in 6 ounces of pumpkin is 61.5 calories.
What is the unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Given that, 8 ounces of canned pumpkin has 82 calories
Now, number of calories in 1 ounce = Total number of calories ÷ Number of ounces
= 82/8
= 10.25 calories
Number of calories in 6 ounces of pumpkin
= 6×10.25
= 61.5 calories
Therefore, number of calories in 6 ounces of pumpkin is 61.5 calories.
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Jay is letting her bread dough rise. After three hours, her bread dough is \dfrac{11}{5} 5 11 start fraction, 11, divided by, 5, end fraction of its original size.
Answer:
Jay's bread size is 220% of the original size.
Step-by-step explanation:
The question is incomplete.
The complete question is as follows.
Jay is letting her bread rise. After 3 hours,her bread is at 11/5 of its original size. What percent of its original size is jays bread dough?
Solution:
Let the original bread size be = [tex]100[/tex] units
After 3 hours the bread rises = [tex]\frac{11}{5}[/tex] of the original size
New size of bread = [tex]\frac{11}{5}\times 100=220[/tex] units
Percent of original size the new bread is
⇒ [tex]\frac{New\ bread\ size}{Original\ bread\ size}\times 100[/tex]
⇒ [tex]\frac{220}{100}\times100[/tex]
⇒ [tex]220\%[/tex]
Ben paid 45 dollers for an old giutar than resold it after marking up the price 15 persent he used 36 persent of the money he got fornthe giutar tpo buy books how much money did ben spend on books
Answer:
18.63
Step-by-step explanation:
45*115%= 45*1.15 = 51.75
51.75*0.36 = 18.63
Answer:
18.63 dollars.
Step-by-step explanation:
The selling price was 45 + 15% or 45
= 45 + 0.15*45
= 51.75 dollars.
36% of 51.75
= 0.36 * 51.75
= 18.63 dollars which he had to spend on books.
triangle ABC and triangle DEF are drawn below. If AB congruent DE, AC congruent DF and angle A congruent to angle D write a seuence of transfornation that maps tiangle ABC onto triangle DEF
Answer:
First you would translate triangle ABC to the right . next you would then translate triangle ABC up . Last you would rotate triangle ABC clockwise and matched angle A with angle D.
Step-by-step explanation:
Given that lines a and b are parallel, what angles formed on line b when cut by the transversal are congruent with ∠2?
Answer:
The angles formed on line b when cut by the transversal are congruent with ∠2 are [tex]\angle{6}\text{ and }\angle{7}[/tex]
Step-by-step explanation:
Consider the provided information.
If transversal line crossed by two parallel lines, then, the corresponding angles and alternate angles are equal .
The angles on the same corners are called corresponding angle.
Alternate Angles: Angles that are in opposite positions relative to a transversal intersecting two lines.
∠2 and ∠6 are corresponding angles
Therefore, ∠2 = ∠6
∠2 and ∠7 are alternate exterior angles
Therefore, ∠2 = ∠7
Hence, the angles formed on line b when cut by the transversal are congruent with ∠2 are [tex]\angle{6}\text{ and }\angle{7}[/tex]
Simplify.
7/8+(−2/3) divided by 5/6
Enter your answer, in simplest form, in the boxes.
Answer:
your answer is 1/4
Step-by-step explanation:
part 1
7/8 + -2/3 = 5/24
part 2
5/24÷5/6 = 1/4
A 500-turn circular coil with an area of 0.050 m^2 is mounted on a rotating frame, which turns at a rate of 20.0 rad/s in the presence of a 0.050-T uniform magnetic field that is perpendicular to the axis of rotation. What is the instantaneous emf in the coil at the moment that the normal to its plane is at a 30.0 degree angle to the field?
a) zero
b) 12.5 V
c) 21.6 V
d) 25.0 V
Answer:
option (b) 12.5 V
Step-by-step explanation:
Given:
Number of turns, N = 500
Area of the coil, A = 0.050 m²
Angular speed, ω = 20 rad/s
Magnetic field, B = 0.050 T
Angle to the field, θ = 30°
Now,
EMF induced, ε = NBAωsinθ
on substituting the values, we get
ε = 500 × 0.050 × 0.050 × 20 × sin30°
or
ε = 25 × 0.5
or
ε = 12.5 V
Hence,
option (b) 12.5 V
answer correcly / explain a lil.
match them....
Answer:
[tex]\displaystyle \boxed{66 \times 7}\:7(60 + 6)[/tex]
[tex]\displaystyle \boxed{97 \times 4}\:4(100 - 3)[/tex]
[tex]\displaystyle \boxed{8(4 + 2)}\:32 + 16 = 48[/tex]
[tex]\displaystyle \boxed{5(9 - 6)}\:(5 \times 9) - (5 \times 6)[/tex]
[tex]\displaystyle \boxed{3(4 + 7)}\:(3 \times 4) + (3 \times 7)[/tex]
Step-by-step explanation:
According to the Order of Operations [GEMS\BOMDAS\PEMDAS etc.], you evaluate everything in parentheses first before preceding with your Division & Multiplication and Subtraction & Addition. When you do this, you will know exactly which expression corresponds with its Distributive Property expression.
I am joyous to assist you anytime.
You have a $30 gift card from Amazon that you use to purchase two T-shirts for a total of $25. The sales tax is 8%. Will your gift card cover the cost? Write an expressions and solve. Explain your answer.
Answer:
Step-by-step explanation:
Total cost of the two T-shirts that you purchased is $25. The sales tax on the T-shirts is 8%. This means that the additional amount that you will pay on the T shirt is
25/100 × 25 = 0.25 × 25 = $6.25
The new cost of the T shirt would be the sum of the original cost and the value of the sales tax. Therefore,
New cost = 25 + 6.25 = $31.25
You have a $30 gift card from Amazon. The new cost of the shirts is higher than that value of the gift card. So the gift card will not cover the cost of the T shirts
To determine if the gift card will cover the cost of the two T-shirts, calculate the sales tax and subtract it from the total cost. The gift card will cover the cost if the remaining balance is positive.
Explanation:To determine if your gift card will cover the cost of the two T-shirts, you need to calculate the cost of the sales tax and subtract it from the total cost of the T-shirts. First, convert the sales tax percentage to decimal form by dividing it by 100 (8% becomes 0.08). Then, multiply the total cost of the T-shirts (which is $25) by the sales tax rate to find the amount of sales tax. Multiply $25 by 0.08 to get $2. Next, subtract the sales tax amount from the total cost of the T-shirts: $25 - $2 = $23. Finally, determine if the gift card covers the cost by comparing it to the final cost: $30 - $23 = $7.
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