1) The domain of Y = 3x⁴ + 4x³ is (-∞, ∞).
2) The x-intercepts are (0, 0) and (-4/3, 0) and the y-intercept is (0, 0)
3) The horizontal asymptote is y = infinity.
4) Function does not exhibit any symmetry with respect to the y-axis or origin.
5) The critical points are x = 0 and x = -1.
6) The critical points are x = 0 and x = -1.
7) The function is concave down on the interval (-∞, -2/3) and concave up on the intervals (-2/3, 0) and (0, ∞).
How to find domain?1) The domain of a polynomial function is all real numbers, so the domain of Y = 3x⁴ + 4x³ is (-∞, ∞).
How to find Intercepts?2) To find the x-intercepts, we set Y equal to zero and solve for x:
0 = 3x⁴ + 4x³
0 = x³(3x + 4)
x = 0 or x = -4/3
Therefore, the x-intercepts are (0, 0) and (-4/3, 0).
To find the y-intercept, we set x equal to zero and solve for Y:
Y = 3(0)⁴ + 4(0)³
Y = 0
Therefore, the y-intercept is (0, 0).
How to find Asymptotes?3) Polynomial functions do not have vertical asymptotes. However, as x approaches positive or negative infinity, the function approaches infinity. Therefore, the horizontal asymptote is y = infinity.
How to find Symmetry?4) The function Y = 3x⁴ + 4x³ is neither even nor odd. Therefore, it does not exhibit any symmetry with respect to the y-axis or origin.
How to find Critical Points?5) To find the critical points, we take the first derivative of Y and set it equal to zero:
Y' = 12x³ + 12x²
0 = 12x²(x + 1)
Therefore, the critical points are x = 0 and x = -1.
How to find Maxima/Minimum?6) To determine whether the critical points are maxima or minima, we take the second derivative of Y and evaluate it at each critical point:
Y'' = 36x² + 24x
At x = 0, Y'' = 0, which means that the second derivative test is inconclusive. To determine whether x = 0 is a maxima or minima, we look at the sign of the first derivative to the left and right of the critical point. We find that Y' is negative to the left of x = 0 and positive to the right, so x = 0 is a local minimum.
At x = -1, Y'' = 12, which is positive. Therefore, x = -1 is a local minimum.
How to find Concavity?7) To determine the concavity of the function, we look at the sign of the second derivative:
Y'' = 36x² + 24x
When Y'' > 0, the function is concave up, and when Y'' < 0, the function is concave down.
At x < -2/3, Y'' is negative, so the function is concave down.
At -2/3 < x < 0, Y'' is positive, so the function is concave up.
At x > 0, Y'' is positive, so the function is concave up.
Therefore, the function is concave down on the interval (-∞, -2/3) and concave up on the intervals (-2/3, 0) and (0, ∞).
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Find the prime factorization of each number. Record without and then with exponents. 75
Answer:
5*5*3
5²*3
Step-by-step explanation:
Assuming that we have to find prime factorization of 75:
75 = 25 * 3
75 = 5*5*3
PLEASE HELP ME WITH THIS MATH PROBLEM!!! WILL GIVE BRAINLIEST!!! 20 POINTS!!!
The average price of a gallon of milk in the following years, using the exponential growth function, are:
a) 2018 = $2.90
2021 = $3.55
b) Based on the exponential growth function, the cost of milk is inflating at 7% per year.
c) Based on the percentage of inflation, the predicted price of a gallon of milk in 2025 is $4.66.
What is an exponential growth function?An exponential growth function is a mathematical equation that describes the relationship between two variables (dependent and independent).
Under the function, there is a constant ratio of growth with the number of years between the initial value and the desired value as the exponent.
The given function for the price of average gallon of milk from 2008 to 2021 is 3.55 = 2.90 (1 + x)³.
Average price of milk in 2018 = $2.90
Average price of milk in 2021 = $3.55
Change in the average price of milk = $0.65 ($3.55 - $2.90)
The percentage change from 2018 to 2021 = 22.41% ($0.65 ÷ $2.90 x 100)
The cost of milk is inflating annually at (1 + x)^3
x = 7%
Cost of milk in 2025 = y
Number of years from 2018 to 2025 = 7 years
y = 2.90 (1 + 0.07)⁷
y = 2.90 (1.07)⁷
y = $4.66
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the produce manager at the local pig & whistle grocery store must determine how many pounds of
bananas to order weekly. based upon past experience, the demand for bananas is expected to be 100,
150, 200, or 250 pounds with the following probabilities: 100lbs 0.20; 150lbs 0.25, 200lbs 0.35, 250lbs 0.20.
the bananas cost the store $.45 per pound and are sold for $.085 per pound. any unsold bananas at the
end of each week are sold to a local zoo for $.30 per pound. use your knowledge of decision analysis to
model and solve this problem in order to recommend how many pounds of bananas the manager should
order each week
As per the probability, the expected demand for bananas per week is 182.5 pounds.
To model this problem, we can use decision analysis, which involves identifying the possible outcomes, assigning probabilities to each outcome, and calculating the expected value of each decision.
In this case, the possible outcomes are the demand for bananas, which can be 100, 150, 200, or 250 pounds per week. The probabilities of each demand level are given as 0.20, 0.25, 0.35, and 0.20, respectively.
Let X denote the demand for bananas in pounds. Then, the expected demand for bananas, denoted as E(X), can be calculated as follows:
E(X) = 100(0.20) + 150(0.25) + 200(0.35) + 250(0.20) = 182.5
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A paving company is paving the rectangular student parking lot at hamilton high school. the length of the parking lot can be represented as (5 − 7) and the width as (3 + 4) . which expression represents the area of the parking lot?
The expression that represents the area of the parking lot is (5 - 7) * (3 + 4).
How to express the area of the parking lot?The length of the parking lot can be represented as (5 - 7), which simplifies to -2. The width of the parking lot can be represented as (3 + 4), which simplifies to 7. To find the area of a rectangle, we multiply the length by the width.
Therefore, the expression that represents the area of the parking lot is (-2) * 7. Multiplying -2 by 7 gives us -14. So, the area of the parking lot is -14 square units.
However, it is important to note that negative areas do not have practical meaning in this context, as areas are typically positive values. Therefore, the area of the parking lot would be considered 14 square units.
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A social scientist would like to analyze the relationship between educational attainment (in years of higher education) and annual salary (in $1,000s). He collects data on 20 individuals. A portion of the data is as follows: Salary Education 40 3 53 4 ⋮ ⋮ 38 0 Click here for the Excel Data File a. Find the sample regression equation for the model: Salary = β0 + β1Education + ε. (Round answers to 2 decimal places. ) Salaryˆ= + Education b. Interpret the coefficient for Education. Multiple choice As Education increases by 1 year, an individual’s annual salary is predicted to decrease by $8,590. As Education increases by 1 year, an individual’s annual salary is predicted to decrease by $10,850. As Education increases by 1 year, an individual’s annual salary is predicted to increase by $8,590. As Education increases by 1 year, an individual’s annual salary is predicted to increase by $10,850. C. What is the predicted salary for an individual who completed 7 years of higher education? (Round coefficient estimates to at least 4 decimal places and final answer to the nearest whole number. ) Salaryˆ $
The sample regression equation for salary and education is Salaryˆ= 32.67 + 4.46Education. For each additional year of education, an individual's salary is predicted to increase by $4,460. Predicted salary for 7 years of education is $63,845.
Using the provided data, we can calculate the sample regression equation for the model Salary = β0 + β1Education + ε by using linear regression. The result is Salaryˆ= 32.67 + 4.46Education.
The coefficient for Education is 4.46, which means that as Education increases by 1 year, an individual’s annual salary is predicted to increase by $4,460.
To find the predicted salary for an individual who completed 7 years of higher education, we substitute Education = 7 into the regression equation: Salaryˆ= 32.67 + 4.46(7) = $63,845. Therefore, the predicted salary for an individual who completed 7 years of higher education is $63,845.
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Jack is making $9.50 per hour at his job re-stocking grocery shelves. His supervisor is impressed with his good work and gives him a 10% raise. How much per hour is Jack making now?
PLS HELP I WILL MARK YOU BRAINLIEST
Answer:
Jack now makes $10.45
Step-by-step explanation:
First, we need to find 10% of 9.50. To do that we must divide 9.50 by 10 or simply move the decimal one place to the left to get 0.95. 0.95 is 10% of 9.50. Now we must ADD 0.95 to 9.50 because Jack is getting a 10% raise. 0.95 + 9.50 = 10.45. Therefore Jack makes $10.45 per hour at his job.
(−3m
5
)(−2m
4
)=left parenthesis, minus, 3, m, start superscript, 5, end superscript, right parenthesis, left parenthesis, minus, 2, m, start superscript, 4, end superscript, right parenthesis, equals
The solution to the equation is m = 0.
What is an algebraic expression?
An algebraic expression is a mathematical phrase that contains variables, constants, and mathematical operations. It may also include exponents and/or roots. Algebraic expressions are used to represent quantities and relationships between quantities in mathematical situations, often in the context of problem-solving.
To solve the equation:
(-3m^ {5}) (-2m^ {4}) = (-6m^ {9})
We can simplify the left side of the equation by multiplying the terms:
(-3m^ {5}) (-2m^ {4}) = (6m^ {9})
Now we have:
6m^ {9} = (-6m^ {9})
To solve for m, we can divide both sides by 6m^ {9}:
m^ {9} = -m^ {9}
Since the powers of m on both sides are equal, we can simplify to:
2m^ {9} =0
Dividing both sides by 2, we get:
m^ {9} =0
Taking the ninth root of both sides, we get:
m = 0
Therefore, the solution to the equation is m = 0.
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Complete Question:
Simplify the expression: $(-3m^ {5}) (-2m^ {4}) $.
Explain why a runner completes a 6.2 mi race in 32 min , then he must be running 11m/hr at the entire race
The runner's average speed is approximately 11.63 mph, which is slightly faster than 11 mph. So, the runner doesn't maintain exactly 11 mph throughout the entire race, but they are close.
If a runner completes a 6.2 mile race in 32 minutes, we can calculate their average speed by dividing the total distance by the total time.
6.2 miles ÷ 32 minutes = 0.194 miles per minute
To convert this to miles per hour, we need to multiply by 60 (the number of minutes in an hour):
0.194 miles per minute x 60 minutes = 11.63 miles per hour
So the runner must be running at an average speed of 11.64 miles per hour throughout the entire race in order to complete it in 32 minutes. This is an impressive pace and shows that the runner is very fit and capable of sustaining a fast speed for a relatively long distance.
A runner completes a 6.2-mile race in 32 minutes, and we are to determine if they maintain an 11 mph pace throughout the race. To do this, we need to convert the race time to hours and then divide the race distance by the time.
First, convert 32 minutes to hours:
32 minutes / 60 minutes/hour = 0.5333 hours
Next, calculate the average speed:
6.2 miles / 0.5333 hours ≈ 11.63 mph
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A speaker is in the shape of a cube with an edge length of 3. 5 inches. The speaker is sold in a package in the shape of a square prism with a base area of 16 square inches and a height of 4. 25 inches. How much empty space, in cubic inches, remains in the package after the speaker is placed in the package?
The volume of the cube-shaped speaker is 42.875 cubic inches. The volume of the package is 68 cubic inches. Subtracting the volume of the speaker from the volume of the package gives the empty space remaining, which is 25.125 cubic inches.
The volume of the cube-shaped speaker is given by V₁ = (edge length)³ = 3.5³ = 42.875 cubic inches.
The volume of the square prism-shaped package is given by V₂ = (base area) x (height) = 16 x 4.25 = 68 cubic inches.
Therefore, the empty space remaining in the package after the speaker is placed in it is V₂ - V₁ = 68 - 42.875 = 25.125 cubic inches.
So, the empty space remaining in the package is 25.125 cubic inches.
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Thirty-three cities were researched to determine whether they had a professional sports team, a symphony, or a children's museum. Of these cities, 17 had a professional sports team, 15 had a symphony, 14 had a children's museum, 9 had a professional sports team and a symphony, 6 had a professional sports team and a children's museum, 6 had a symphony and a children's museum, and 3 had all three activities.
Complete parts a) through e) below.
a) How many of the cities surveyed had only a professional sports team?
b) How many of the cities surveyed had a professional sports team and a symphony, but not a children's museum?
c) How many of the cities surveyed had a professional sports team or a symphony?
d) How many of the cities surveyed had a professional sports team or a symphony, but not a children's museum?
e) How many of the cities surveyed had exactly two of the activities?
Simplify your answers.
a) The number of cities that had only a professional sports team can be found by subtracting the number of cities that had a professional sports team and a symphony, the number of cities that had a professional sports team and a children's museum, and the number of cities that had all three activities from the total number of cities:
33 - (9 + 6 + 3) = 15 cities had only a professional sports team.
b) The number of cities that had a professional sports team and a symphony, but not a children's museum can be found by subtracting the number of cities that had all three activities from the number of cities that had a professional sports team and a symphony:
9 - 3 = 6 cities had a professional sports team and a symphony, but not a children's museum.
c) The number of cities that had a professional sports team or a symphony can be found by adding the number of cities that had a professional sports team, the number of cities that had a symphony, and then subtracting the number of cities that had both:
17 + 15 - 9 + 14 - 6 + 3 = 34 cities had a professional sports team or a symphony.
d) The number of cities that had a professional sports team or a symphony, but not a children's museum can be found by subtracting the number of cities that had all three activities from the answer to part c:
34 - 3 = 31 cities had a professional sports team or a symphony, but not a children's museum.
e) The number of cities that had exactly two of the activities can be found by adding up the number of cities that had a professional sports team and a symphony, the number of cities that had a professional sports team and a children's museum, and the number of cities that had a symphony and a children's museum, and then subtracting twice the number of cities that had all three activities:
9 + 6 + 6 - 2(3) = 15 cities had exactly two of the activities.
Compound Interest:
In March 2003, Natalie invested $800 in an account that earns 4. 8% interest compounded monthly. After 5 years, she withdrew all the money and reinvested it in a new account that earns 6% interest compounded semiannually. Assuming there were no other deposits or withdrawals, how much total interest will she have earned by March 2025?
I NEED HELP, CAN SOMEONE HELP ME, PLEASE?
Natalie will have earned a total of $488.97 in interest by March 2025.
"What is compound interest formula?To solve this problem, we can use the formula for compound interest:
A = [tex]P(1 + r/n)^(nt)[/tex]
where A is the total amount, P is the principal amount, r is the annual interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the time in years.
First, let's find out how much money Natalie will have in her first account after 5 years:
P = $800
r = 4.8% per year = 0.048
n = 12 (compounded monthly)
t = 5 years
A = [tex]800(1 + 0.048/12)^(12*5)[/tex]
A = $995.08
So after 5 years, Natalie will have $995.08 in her first account.
Next, let's find out how much money Natalie will have in her second account:
P = $995.08
r = 6% per year = 0.06
n = 2 (compounded semiannually)
t = 5 years
A = [tex]995.08(1 + 0.06/2)^(2*5)[/tex]
A = $1,288.97
So after reinvesting her money in the second account, Natalie will have $1,288.97 after 5 years.
Finally, let's calculate the total interest earned:
Total interest = A - P
Total interest = $1,288.97 - $800
Total interest = $488.97
Therefore, Natalie will have earned a total of $488.97 in interest by March 2025.
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Evaluate (8 + 2)^3 - 6
Step-by-step explanation:
First you need to do of bracket
(8+2)
=10
Second you need to do of exponential sign ^
10^3=1000(note this sign is also called cube)
Now,
1000-6
=994
Answer:
994
Step-by-step explanation:
We need to use order of operations to solve this problem.
( 8 + 2 ) ^ 3 - 6
According to PEMDAS, Parenthesis must be resolved first.
So we get:
10 ^ 3 - 6
Secondly, PEMDAS states that Exponents go next
1000 - 6
Finally, we only have one operation left, subtraction, so we can go ahead and do that.
994 is your answer.
I put a lot of thought and effort into my answers, so a brainliest would be much appreciated!
The scale drawing shown represents a circular playground with a scale factor of . = . What is the actual area of the playground? Give your answer in terms of pi
The calculated value of the actual area of the playground is 5625π
What is the actual area of the playground?From the question, we have the following parameters that can be used in our computation:
A scale drawing of a playground had a scale of 15
This means that
Scale factor = 15/1
Evaluate
Scale factor = 15
The actual area of the park in meters squared is calculated as
Area = Area of scale * Scale factor²
Substitute the known values in the above equation, so, we have the following representation
Area = Area of scale * 15²
Using the area of circle, we have
Area = π5² * 15²
Evaluate
Area = 5625π
Hence, the actual area is 5625π
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In 2000, 1500 rabbits live in a warren in a certain area. the number of rabbits increases exponentially at a discrete rate of 7% per year. predict population in 2008&2022
The predicted populations of rabbits in the warren in 2008 and 2022 are 2909 and 9933, respectively.
To predict the population of rabbits in the warren in 2008 and 2022, we can use the formula for exponential growth:
P(t) = P₀ (1 + r)ᵗ
Where P(t) is the population at time t, P₀ is the initial population, r is the growth rate, and t is the time elapsed.
For this problem, we know that P₀ = 1500, r = 0.07 (since the growth rate is 7%), and we want to find P(8) and P(22) (since we're predicting the population in 2008 and 2022, respectively).
Using the formula, we get:
[tex]P(8) = 1500( 1 + 0.07)^{8} = 2909[/tex]
So we can predict that there will be about 2909 rabbits in the warren in 2008.
To find P(22), we simply plug in t = 22:
[tex]P(22) = 1500 (1 + 0.07)^{22} = 9933[/tex]
So we can predict that there will be about 9933 rabbits in the warren in 2022.
Therefore, the predicted populations of rabbits in the warren in 2008 and 2022 are 2909 and 9933, respectively.
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The physics department of a college has 10 male professors, 7 female professors, 13 male teaching assistants, and 9 female teaching assistants. If a person is selected at random from the group, find the probability that the selected person is a teaching assistant or a female. (Input you answer as a decimal to four decimal places. )
The probability that the selected person is a teaching assistant or a female is 29.
What is the probability?
A probability is a numerical representation of the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.
Here, we have
Given: The physics department of a college has 10 male professors, 7 female professors, 13 male teaching assistants, and 9 female teaching assistants.
We have to find the probability that the selected person is a teaching assistant or a female.
N(A OR B) = N(A) + N(B) - N(A AND B)
N(teaching assistant) = 13 + 9 = 22
N(females) = 7 + 9 = 16
N(teaching assistant and female) = 9
N(teaching assistant OR female) = N(teaching assistant) + N(females) - N(teaching assistant AND female)
N(teaching assistant OR female) = 22 + 16 - 9
N(teaching assistant OR female) = 29
Hence, the probability that the selected person is a teaching assistant or a female is 29.
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Solve the following problems:
given: circle k(o), diameter us, mru=50°, mut=30°
find: m
The measure of angle M is 20°.
To solve the problem, we need to find the measure of angle M, given the information about Circle K with center O, diameter US, angle MRU = 50°, and angle MUT = 30°.
Step 1: Determine the relationship between angles MRU and MUT.
Since MRU and MUT are both inscribed angles in Circle K, they share the same intercepted arc, which is arc MU.
Step 2: Calculate the measure of arc MU.
The measure of an intercepted arc is twice the measure of the inscribed angle. Since angle MRU = 50°, the measure of arc MU will be 2 * 50° = 100°.
Step 3: Find the measure of angle M.
We know that angle MUT = 30°, and the measure of an intercepted arc is twice the measure of the inscribed angle. Therefore, the measure of arc MT = 2 * 30° = 60°. Now, since arc MU = 100°, we can determine the measure of arc MS (arc MS = arc MU - arc MT) which is 100° - 60° = 40°.
Step 4: Calculate the measure of angle M.
Finally, the measure of angle M can be found using the intercepted arc MS. Since the measure of an intercepted arc is twice the measure of the inscribed angle, angle M = 1/2 * arc MS = 1/2 * 40° = 20°.
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Use the information to answer the question.
A company had a profit of -$4,758 in January and a profit of $3,642 in February. The company's profits for the months of March through May
were the same in each of these months. By the end of May, the company's total profits for the year were -$1,275.
What were the company's profits each month from March through May? Enter the answer in the box.
The company's profits for each month from March through May were $658.33.
How to find the company profits for the months of March through May?
To solve the problem, we need to use the information given and set up an equation. Let's call the profits for the months of March through May "P" (since they are the same for each month).
The company's total profits for the year can be calculated by adding up the profits for each month:
January profit + February profit + March profit + April profit + May profit = total profit
Plugging in the numbers we know:
-$4,758 + $3,642 + 3P + 3P + 3P = -$1,275
Simplifying the equation:
9P = $5,925
Dividing both sides by 9:
P = $658.33
So the company's profits for each month from March through May were $658.33.
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Do not answer 7 and 9
Answer:
[tex]32 \div 4 = 8[/tex]
Answer: 32 divided by 4 =8
Step-by-step explanation:
What is 524/125 written as a decimal?
Answer:
C)4.192
Step-by-step explanation:
What is 524/125 written as a decimal?
We Take
524 divided by 125 = 4.192
So, the answer is C)4.192
At a music store in New York City, 62 people entering the store were selected at random and were asked to choose their favorite type of music. Of the 62, 14 chose rock, 16 chose country, 8 chose classical and 24 chose something other than rock, country, or classical.
a) Determine the empinical probability that the next person entering the store favors rock music.
b) Determine the empirical probability that the next person entering the store favors country music.
c) Determine the empirical probability that the next person entering the store favors something other than rock, country, or classical music
(a) The empirical probability that the next person favors rock music is approximately 0.226.
b) The empirical probability that the next person favors country music is approximately 0.258.
c) The empirical probability that the next person favors something other than rock, country, or classical music is approximately 0.387.
a) To determine the empirical probability that the next person entering the store favors rock music, you need to divide the number of people who chose rock (14) by the total number of people surveyed (62).
Empirical Probability (Rock) = Number of Rock Fans / Total Surveyed = 14 / 62 ≈ 0.226
b) To determine the empirical probability that the next person entering the store favors country music, you need to divide the number of people who chose country (16) by the total number of people surveyed (62).
Empirical Probability (Country) = Number of Country Fans / Total Surveyed = 16 / 62 ≈ 0.258
c) To determine the empirical probability that the next person entering the store favors something other than rock, country, or classical music, you need to divide the number of people who chose something other (24) by the total number of people surveyed (62).
Empirical Probability (Other) = Number of Other Fans / Total Surveyed = 24 / 62 ≈ 0.387
In summary:
a) The empirical probability that the next person favors rock music is approximately 0.226.
b) The empirical probability that the next person favors country music is approximately 0.258.
c) The empirical probability that the next person favors something other than rock, country, or classical music is approximately 0.387.
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Larry went to Home Depot and buck 32 ft.² of treated plywood for $50 and 40 ft.² a regular plywood for $64 how much more does the treated plywood cost in the regular plywood in dollars per foot 
If Larry went to Home Depot and buck 32 ft.² The amount the treated plywood cost in the regular plywood in dollars per foot is: -$1.80 per foot
What is the cost?Treated plywood cost per square foot:
50 / 32
= $1.5625 per square foot
Regular plywood cost per square foot:
64 / 40
= -$1.60 per square foot
Difference in cost per square foot
1.5625 - 1.60
= -$0.0375 per square foot
Difference in cost per foot is:
(-$0.0375 / 0.0208)
≈ $1.80 per foot
Therefore based on the above calculation it treated plywood costs $1.80 less per foot than the regular plywood.
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Evaluate the following indefinite ∫x tan^2 x dx
To evaluate the indefinite integral ∫x tan^2 x dx, we can use integration by parts.
Let u = x and dv = tan^2 x dx. Then, du/dx = 1 and v = ∫tan^2 x dx = tan x - x.
Using the formula for integration by parts, we have:
∫x tan^2 x dx = uv - ∫v du/dx dx
= x(tan x - x) - ∫(tan x - x) dx
= x(tan x - x) + ln|cos x| + C
Therefore, the indefinite integral of x tan^2 x dx is x(tan x - x) + ln|cos x| + C, where C is the constant of integration.
To evaluate the indefinite integral ∫x tan^2(x) dx, we can use integration by parts, which is defined as ∫udv = uv - ∫vdu.
First, let's choose our u and dv:
u = x, so du = dx
dv = tan^2(x) dx
To find v, we need to integrate dv. Since tan^2(x) = sec^2(x) - 1, we get:
v = ∫(sec^2(x) - 1) dx = tan(x) - x
Now, using integration by parts:
∫x tan^2(x) dx = x(tan(x) - x) - ∫(tan(x) - x) dx
Let's evaluate the remaining integral:
∫(tan(x) - x) dx = ∫tan(x) dx - ∫x dx
= ln|sec(x)| - (1/2)x^2 + C
So, the final answer is:
∫x tan^2(x) dx = x(tan(x) - x) - [ln|sec(x)| - (1/2)x^2] + C
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stys
ACA
2. A square with one side length represented by an
expression is shown below.
6(3x + 8) + 32 + 12x
Use the properties of operations to write three
different equivalent expressions to represent the
lengths of the other three sides of the square. One
of your expressions should contain only two terms.
We want to use properties to write expressions for the length of the other sides of the square.
Remember that the length of all the sides in a square is the same, so we only need to rewrite the above expression in two different ways.
First, we can use the distribute property in the first term:
[tex]\sf 6\times(3x + 8) + 32 + 12\times x[/tex]
[tex]\sf = 6\times3x + 6\times8 + 32 +12\times x[/tex]
[tex]= \sf 18\times x + 48 + 32 + 12\times x[/tex]
So this can be the length of one of the sides.
Now we can keep simplifying the above equation:
[tex]= \sf 18\times x + 48 + 32 + 12\times x[/tex]
To do it, we can use the distributive and associative property in the next way:
[tex]\sf 18\times x + 48 + 32 + 12\times x[/tex]
[tex]= \sf 18\times x + 12\times x + 48 + 32[/tex]
[tex]= \sf (18\times x + 12\times x) + (48 + 32)[/tex]
[tex]= \sf (18 + 12)\times x + 80[/tex]
[tex]= \sf 30\times x + 80[/tex]
This can be the expression to the other side.
Parallel lines EN, BH, and RK, with transversal PW are shown. m/BMV-108 and m/KVS-72.
Part A: Based on the diagram above and the given information, what is the
m/WPN?
The value of measure of angle WPN is,
⇒ m ∠WPN = 108°
We have to given that;
Parallel lines EN, BH, and RK, with transversal PW are shown.
And, m ∠ BMV = 108° and m ∠KVS = 72°
Now, We get by definition of vertically opposite angle;
m ∠ BMV = m ∠PWN = 108°
Hence, By definition of alternate angle we get;
⇒ m ∠PWN = m ∠WPN = 108°
Thus, The value of measure of angle WPN is,
⇒ m ∠WPN = 108°
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If a cup of coffee has temperature 95 C in a room where theremperature is 20 C, then, according to Newon's Law of Cooling, thetemperature of the coffee after t minutes is T(t) = 20+ 75e-t/50. What is the average temperature of thecoffee during the first half hour?
To find the average temperature of the coffee during the first half hour, we need to find the temperature of the coffee at t = 0 (when the coffee is just brewed) and at t = 30 (after half an hour has passed).
At t = 0, T(0) = 20 + 75e^0/50 = 20 + 75 = 95 C. At t = 30, T(30) = 20 + 75e^-30/50 ≈ 42.5 C.
So, the temperature of the coffee decreases from 95 C to 42.5 C during the first half hour.
The average temperature during this time period can be found by taking the average of the initial and final temperatures:
Average temperature = (95 C + 42.5 C) / 2 = 68.75 C.
Therefore, the average temperature of the coffee during the first half hour is 68.75 C.
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A bowl contains four balls numbered 1 through 4. You select the four balls successively,
without replacement, until none remain in the bowl.
a) What is the probability that the two even numbered balls are selected before the odd
numbered balls?
b) What is the probability that the balls are not selected in size order (i. E. 1, 2, 3, 4 or 4, 3, 2,1) ?
a) The probability of selecting two even numbered balls before odd numbered balls is 1/6.
b) The probability of not selecting balls in size order is 22/24 or 11/12.
a) There are 4! (24) ways to arrange the four balls. There are 2! ways (2) to arrange the even balls and 2! ways (2) to arrange the odd balls, so there are 2x2=4 favorable ways (2,4,1,3 and 4,2,1,3). The probability is 4/24, which simplifies to 1/6.
b) There are only 2 ways to select balls in size order (1,2,3,4 and 4,3,2,1). Subtracting these from the total arrangements (24-2) results in 22 non-size ordered selections. The probability is 22/24, which simplifies to 11/12.
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(a)
The masses of two animals at a zoo are described, where band care integers.
•The mass of an African elephant is 6, 125,000 grams, or about 6 x 10 grams.
• The mass of a silverback gorilla is 185, 000 grams, or about 2 x 10 grams.
What are the values of b and c?
bu
CH
(b) Part B
Using these estimated values, the mass of the African elephant is about 3 x 10 times the mass of the silverback gorilla, where m is an integer.
What is the value of m?
m
With the masses, the value of a and b will be 6 and 5.
The value of m is 6.
How to calculate the valueThe mass of an African elephant is 6,125,000 grams, or about 6 x 10⁶grams. Thus, b = 6.
The mass of a silverback gorilla is 185,000 grams, or about 1.85 x 10⁵grams. Thus, c = 5.
We are told that the mass of the African elephant is about 10 times the mass of the silverback gorilla, where m is an integer.
Let's write this as an equation:
6 x 10ⁿ = 10(1.85 x 10⁵)
Simplifying this equation, we get:
6 x 10ⁿ = 1.85 x 10⁶
10ⁿ = 3.08 x 10⁵
Taking the logarithm (base 10) of both sides, we get:
m = log(3.08 x 10)
Using a calculator, we find that:
m ≈ 5.49
Since m must be an integer, we round up to the nearest integer and get:
m = 6.
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Find all solutions of the equation in radians.
sin(2x)sin(x)+cos(x)=0
Answer:
[tex]x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n[/tex]
Step-by-step explanation:
Given equation:
[tex]\sin(2x)\sin(x)+\cos(x)=0[/tex]
Rewrite sin(2x) using the trigonometric identity sin(2x) = 2sin(x)cos(x):
[tex]\implies 2\sin(x)\cos(x)\sin(x)+\cos(x)=0[/tex]
[tex]\implies 2\sin^2(x)\cos(x)+\cos(x)=0[/tex]
Factor out cos(x):
[tex]\implies \cos(x)\left[2\sin^2(x)+1\right]=0[/tex]
Applying the zero-product property:
[tex]\textsf{Equation 1:}\quad\cos(x)=0[/tex]
[tex]\textsf{Equation 2:}\quad2\sin^2(x)+1=0[/tex]
Solve each part separately.
[tex]\underline{\sf Equation \; 1}[/tex]
[tex]\begin{aligned}\cos(x)&=0\\x&=\arccos(0)\\x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n\end{aligned}[/tex]
[tex]\underline{\sf Equation \; 2}[/tex]
[tex]\begin{aligned}2\sin^2(x)+1&=0\\\sin^2(x)&=-\dfrac{1}{2}\;\;\;\;\;\;\leftarrow\;\textsf{No solution}\end{aligned}[/tex]
Therefore, the solutions of the equation in radians are:
[tex]\boxed{x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n}[/tex]
I absolutely hate IQR so can someone help
The interquartile range (IQR) of the given data set is 4.
Interquartile range (IQR) is a measure of variability in a data set that measures the spread of the middle 50% of the data. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1).
How to fine the interquartile range (IQR)?To find the interquartile range (IQR), we first need to find the median of the data set.
The median is the middle value of the dataset when the data is arranged in order. In this case, the data set is already in order, so the median is the middle value or the average of the two middle values:
Median = (26 + 28) / 2 = 27
Now, we need to find the first quartile (Q1) and the third quartile (Q3) of the data set.
Q1 is the median of the lower half of the data set, and Q3 is the median of the upper half of the data set.
Lower half: 22, 24, 26
Upper half: 28, 30
Q1 = median of the lower half = (24 + 26) / 2 = 25
Q3 = median of the upper half = (28 + 30) / 2 = 29
Now we can find the IQR:
IQR = Q3 - Q1 = 29 - 25 = 4
Therefore, the interquartile range (IQR) of the given data set is 4.
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Jasmine creates a map of her town on the coordinate plane. The unit on the coordinate plane is one block.
The locations of the school, post office, and library are given. school (-4,1)
post office (2,1)
library (2,-4)
Move the points of each building to its correct location on the coordinate plane. Jasmine walks from the school to the post office and then to the library.
What is the total distance, in blocks, of her walk?
Jasmine walks from the school to the post office, which is a distance of $2 - (-4) = 6$ blocks horizontally and 0 blocks vertically, so the distance is 6 blocks. Then she walks from the post office to the library, which is a distance of $2 - 2 = 0$ blocks horizontally and $-4 - 1 = -5$ blocks vertically, so the distance is 5 blocks.
The total distance of Jasmine's walk is the sum of the distances of each leg of her journey, which is $6 + 5 = 11$ blocks. Therefore, Jasmine walks 11 blocks in total.