A system of linear inequalities is shown
below. Which of the following points are
solutions of the system?
8x + 5y > 40
-6x +2y >_-18
Answers
The points (0,10) and (-8,4) are solutions of the system of inequalities.
Define system of inequalityA system of inequalities refers to a set of two or more inequalities that are graphed together on the same coordinate plane. Each inequality in the system represents a region in the plane where the solution to that inequality lies.
To determine if a point is a solution of the system of inequalities, we need to check if it satisfies both inequalities. Let's check each point given:
(4,6):
8(4) + 5(6) = 47 > 40 (true)
-6(4) + 2(6) = -14 ≥ 18 (false)
(9,8):
8(9) + 5(8) = 97 > 40 (true)
-6(9) + 2(8) = -40 ≥ 18 (false)
(0,10):
8(0) + 5(10) = 50 > 40 (true)
-6(0) + 2(10) = 20 ≥ 18 (true)
(5,-6):
8(5) + 5(-6) = 34 > 40 (false)
-6(5) + 2(-6) = -30 ≥ 18 (false)
(-1,-8):
8(-1) + 5(-8) = -44 ≤ 40 (false)
-6(-1) + 2(-8) = 10 ≥ 18 (false)
(-8,4):
8(-8) + 5(4) = -24 ≤ 40 (true)
-6(-8) + 2(4) = 52 ≥ 18 (true)
Therefore, the points (0,10) and (-8,4) are solutions of the system of inequalities.
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Related Questions
Kimberly rolls two six-sided number cubes numbered 1 through 6 and adds up the two numbers construct a tree diagram to determine all the possible outcomes list the sum at the end of each branch of the tree
Answers
When Kimberly rolls two six-sided number cubes numbered 1 through 6, it creates 36 possible outcomes which is represent in the tree diagram below
What is a tree diagram?A tree diagram is a visual representation of outcomes. It consists of branches that represent the possible outcomes of each step.
When it comes to Kimberly rolling two six-sided number cubes, we can start by rolling the first cube, and then rolling the second cube.
For each roll of the first cube, there are six possible outcomes (1 to 6). For each outcome of the first cube, there are six possible outcomes for the second cube.
This results in a total of 6 x 6 = 36 possible outcomes.
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Find the range and mean of each data set. Use your results to compare the two data sets.
Set A:
1 10 7 17 20
Set B:
10 17 16 18 12
Answers
Answer:
Set A: 1, 7, 10, 17, 20
Range: 19
Mean: 11
Set B: 10, 12, 16, 17, 18
Range: 8
Mean: 14.6
How to find...
Mean: Divide the sum of all values in a data set by the number of values.
Range: Find the largest observed value of a variable (the maximum) and subtract the smallest observed value (the minimum).
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Select the needed observations and steps before you can factor a difference of two squares.
binomial
trinomial
multiply factors
two negative
prime
two positive
look for a gcf
one positive
one negative
( its a multi choice question )
Answers
In factorization the needed observations and steps before you can factor a difference of two squares are binomial, two positive/negative, prime, multiply factors, and look for a GCF.
Finding the factors of a given number or statement is the process of factorization. Factorization is the process of taking a larger number or expression and turning it into a product of smaller numbers or expressions, or factors. The factors may be polynomials, integers, or other mathematical constructs.
Therefore, the needed observations and steps before you can factor a difference between two squares are:
Binomial: The expression must be written in the form of a binomial, which calls for two terms (for example, x² - 9).Two phrases that are positive or negative must be separated by a minus sign (-), and each term must be a perfect square. Because x² and 9 are both perfect squares and because 9 is the square of 3, for instance, x² - 9 is a difference of two squares.prime: The words must be prime, which means they can't be factored further.Factoring the difference between two squares involves multiplying and then removing the components of each perfect square.To learn more about Factorization, refer to:
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PLEASE ANSWER QUICKLY FOR THE LOVE OF EVERYTHING
Mrs. Robinson surveyed her class about what flavor cake and ice cream they wanted for their class party. The results were split evenly between the cake with 15 choosing chocolate cake and 15 choosing yellow cake. Of the students who chose chocolate cake, 12 also chose vanilla ice cream. There were 7 students in all that chose strawberry ice cream. Construct a two -way table summarizing the data
Answers
The two-way table is of the class survey is:
Vanilla Ice Cream | Strawberry Ice Cream | Total
Chocolate Cake | 12 | 3 | 15
Yellow Cake | 11 | 4 | 15
Total | 23 | 7 | 30
A two-way table summarizing the data from Mrs. Robinson's class survey on cake and ice cream preferences can be constructed as follows.
1: Create a table with rows for Chocolate Cake and Yellow Cake, and columns for Vanilla Ice Cream, Strawberry Ice Cream, and Total.
2: Fill in the given information:
15 students chose Chocolate Cake and 15 students chose Yellow Cake, so put 15 in the Total column for both rows.12 students who chose Chocolate Cake also chose Vanilla Ice Cream, so put 12 in the intersection of Chocolate Cake and Vanilla Ice Cream.There were 7 students in all that chose Strawberry Ice Cream, so put 7 in the Total row of the Strawberry Ice Cream column.3: Complete the table using the given information:
Since 12 students who chose Chocolate Cake also chose Vanilla Ice Cream, 3 students chose Chocolate Cake and Strawberry Ice Cream (15 total - 12).There are 7 students in total who chose Strawberry Ice Cream, so 4 students chose Yellow Cake and Strawberry Ice Cream (7 total - 3).The remaining 11 students chose Yellow Cake and Vanilla Ice Cream (15 total - 4).So, the completed two-way table is:
Vanilla Ice Cream | Strawberry Ice Cream | Total
Chocolate Cake | 12 | 3 | 15
Yellow Cake | 11 | 4 | 15
Total | 23 | 7 | 30
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An angle measure 94 less than the measure of its supplementary angle. What is the measure of each angle?
Answers
The angle measures 43 degrees and its supplementary angle measures 180 - 43 = 137 degrees.
What is the supplementary angle?In geometry, the supplementary angle of an angle is the angle that, when added to the given angle, results in a sum of 180 degrees. In other words, two angles are supplementary if their sum is 180 degrees.
For example, if an angle measures 60 degrees, its supplementary angle would measure 120 degrees, since 60 degrees + 120 degrees = 180 degrees.
According to the given informationLet x be the measure of the angle in degrees.
By definition, the supplementary angle of x measures 180 - x degrees.
We are given that the angle measures 94 degrees less than its supplementary angle, so we can write:
x = (180 - x) - 94
Simplifying and solving for x, we get:
2x = 180 - 94
2x = 86
x = 43
Therefore, the angle measures 43 degrees and its supplementary angle measures 180 - 43 = 137 degrees.
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compute the critical value z a/2 that corresponds to 97% level of confidence, compute the critical value z a/2 that corresponds to 80% level of confidence?
Answers
The critical value z a/2 that corresponds to a 97% level of confidence is 1.96, and the critical value z a/2 that corresponds to an 80% level of confidence is 1.28.
In statistical hypothesis testing, the critical value is the point beyond which we reject the null hypothesis. To find the critical values, we need to use the standard normal distribution table or calculator.
For a 97% level of confidence, the significance level (α) is 0.03, and the critical value can be found by dividing α by 2 to get 0.015 (since it's a two-tailed test), and then finding the corresponding z-value using the standard normal distribution table or calculator. The z-value is 1.96.
Similarly, for an 80% level of confidence, the significance level (α) is 0.20, and the critical value can be found by dividing α by 2 to get 0.10 (since it's a two-tailed test), and then finding the corresponding z-value using the standard normal distribution table or calculator. The z-value is 1.28.
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In the mid-nineteenth century, explorers used the boiling point of water to estimate altitude. The boiling temperature of water T (in °F) can be approximated by the model T = -1. 83a + 212, where a is the altitude in thousands of feet. Two campers hiking in Colorado boil water for tea. If the water boils at 196°F, approximate the altitude of the campers. Give the result to the nearest hundred feet
Answers
The Colorado hikers are at an altitude of 8753.1694 feet.
Here we have been given the equation T = -1. 83a + 212.
Here, T is the boiling point in Fahrenheit of water, while a is the altitude of the place.
We know that the campers in hiing in Colorado boil water for ea. The water boils at a temperature of 196°F, we need to find the altitude.
Here we are going to clearly take T = 196°F
Substituting the value in the above equation will give us
196 = - 1.83a + 212
Taking the variable to the Right Hand Side will give us
196 + 1.83a = 212
Taking 196 to LHS to separate the constant and variable will give us
1.83a = 212 - 196
Solving LHS gives us
1.83a = 16
Taking 1.83 to the other side will give us
a = 16/1.83
or, a = 8.7431694 thousands feet
= 8753.1694 feet
Hence, the Colorado hikers are at an altitude of 8753.1694 feet.
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Write the polynomial in standard form with roots of 1/4 and +5i
Answers
The polynomial with roots of 1/4 and +5i in standard form is:
f(x) = x² - (1/4)x² + 25x - (25/4)
To write the polynomial with roots of 1/4 and +5i in standard form, we need to use the fact that the roots of a polynomial are related to its factors. Specifically, if r is a root of a polynomial, then x - r is a factor of the polynomial.
Therefore, if the roots of our polynomial are 1/4 and +5i, then we know that the factors of the polynomial are:
(x - 1/4) and (x - 5i) and (x + 5i)
To get the polynomial in standard form, we need to multiply out these factors and simplify.
(x - 1/4) and (x - 5i) and (x + 5i) = (x - 1/4) and (x² - 25i²)
= (x - 1/4) and (x² + 25)
= x³ + 25x - (1/4)x² - (25/4)
Therefore, the polynomial with roots of 1/4 and +5i in standard form is:
f(x) = x³ - (1/4)x² + 25x - (25/4)
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Toad and Toadette just had their first little toadstool! Toad's family is known to be purebred dominant for red spots on their white cap. Everyone was shocked when Little Toad was born with a white cap with white spots instead of red. Toadette is very upset as she thinks the Mushroom Kingdom Hospital accidentally switched babies. Is this true? Did the hospital really switch babies? Choose either "yes" or "no" and defend your answer.
Answers
No, it is not necessarily true that the hospital switched babies.
What happened there?Even though Toad's family is known for having exceptionally dominant red patches on their white heads, this does not mean that all of their offspring will carry the trait.
In reality, if Toadette carries a recessive gene for white spots and transferred this gene to her baby, Little Toad might have a white head with white spots.
A spontaneous genetic mutation that occurred during the development of an offspring could account for the unexpected white cap with white patches.
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To the nearest hundredth, what is the relative frequency of boys who want to go to the water park
Answers
To find the relative frequency of boys who want to go to the water park, we need to divide the number of boys who want to go to the water park by the total number of boys in the sample:
Relative frequency = Number of boys who want to go to the water park / Total number of boys
Assuming you have the necessary data, you can compute this value by dividing the number of boys who want to go to the water park by the total number of boys and rounding the result to the nearest hundredth.
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Find the equation of the tangent line to the curve (a lemniscate) 2(x^2+y^2) = 25 (z^2-y^2) at the point (-3, -1)
Answers
The equation of the tangent line to the lemniscuses 2(x²+y²) = 25 (z²-y²) at the point (-3, -1) is y = (16/25)x + 23/25.
To find the equation of the tangent line to a curve, we need to take the derivative of the equation of the curve and evaluate it at the given point.
First, let's rewrite the equation of the lemniscate in terms of x and y:
2(x² + y²) = 25(z² - y²)
Dividing both sides by 25, we get:
(x² + y²) / (25/2) = (z² - y²) / 12.5
Now, we can take the partial derivatives with respect to x and y:
∂/∂x [(x² + y²) / (25/2)] = (2x) / (25/2) = (4x) / 25
∂/∂y [(x² + y²) / (25/2)] = (2y) / (25/2) = (4y) / 25
Next, we need to find the value of z at the point (-3, -1). To do this, we can substitute x = -3 and y = -1 into the equation of the lemniscate:
2((-3)² + (-1)²) = 25(z² - (-1)²)
20 = 25(z² + 1)
z^2 = 19/25
z = ±sqrt(19)/5
Since we want the tangent line at the point (-3, -1), we'll use z = -sqrt(19)/5.
Now, we can evaluate the partial derivatives at (-3, -1, -sqrt(19)/5):
(4(-3)) / 25 = -12/25
(4(-1)) / 25 = -4/25
So, the slope of the tangent line is:
m = ∂z/∂x × -12/25 + ∂z/∂y × -4/25
m = (2x / (25/2)) × (-12/25) + (2y / (25/2)) × (-4/25)
m = -24x/125 - 8y/125
m = -24(-3)/125 - 8(-1)/125
m = 72/125 + 8/125
m = 80/125
m = 16/25
Finally, we can use the point-slope form of a line to find the equation of the tangent line:
y - (-1) = (16/25)(x - (-3))
y + 1 = (16/25)(x + 3)
y = (16/25)x + 48/25 - 25/25
y = (16/25)x + 23/25
So the equation of the tangent line to the lemniscuses 2(x²+y²) = 25 (z²-y²) at the point (-3, -1) is y = (16/25)x + 23/25.
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Solve for "B" I tried it the way I solved the rest of the equations but I can't get it
Answers
Answer:b=30
Step-by-step explanation:
Alternate angles are the same so the angle opposite 30=125
125+30=155.but angles on a straight line=180
So we subtract 180-155 which leaves us with 25 for the top angle in the triangle.Angles in a triangle must add to 180 so we do, 125+25=150
180-150=30
Gary has a brother and a sister in college. He traveled 2 x 10^3 miles to visit his sister. He traveled 4. 2 x 10^5 miles to visit his brother. The distance Gary traveled to visit his brother is how many times as much as the distance Gary traveled to visit his sister?
Answers
The distance Gary traveled to visit his brother is 2.1 x 10^2 times as much as the distance he traveled to visit his sister.
To determine how many times the distance to visit Gary's brother is compared to the distance to visit his sister, we'll follow these steps:
1. Identify the distances traveled:
- Sister: 2 x 10^3 miles
- Brother: 4.2 x 10^5 miles
2. Divide the distance to the brother by the distance to the sister:
(4.2 x 10^5 miles) / (2 x 10^3 miles)
3. Simplify the expression:
- First, let's divide the coefficients: 4.2 ÷ 2 = 2.1
- Next, divide the exponents: 10^5 ÷ 10^3 = 10^(5-3) = 10^2
4. Combine the results:
2.1 x 10^2
So, the distance Gary traveled to visit his brother is 2.1 x 10^2 times as much as the distance he traveled to visit his sister.
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Here is a list of statistical questions. What data would you collect and analyze to answer each question? For numerical data, include the unit of measurement that you would use.
What is a typical height of female athletes on a team in the most recent international sporting event?
Are most adults in the school football fans?
How long do drivers generally need to wait at a red light in Washington, DC
Answers
You can collect data on the heights of all female athletes on a team using recent international sporting event and This data can be gotten by using official event records or by carrying out a survey of the athletes. Heights are said to be measured in centimeters (cm) or feet and inches (ft, in).
Are most adults in the school football fans?To address this question, you can gather information regarding the quantity of school staff members who are passionate about football and compare it against the full count of school staff members. Gathering this information is possible through either conducting a survey or analyzing the number of tickets sold for school football games. The information can be shown using ratio or a percentage.
How long do drivers generally need to wait at a red light in Washington, DC
One can get information on the durations motorists must wait at various traffic signals in Washington, DC. One can ask this information by either installing sensors or cameras at different intersections or by gathering input from drivers through a survey. The duration of waiting periods is commonly quantified in units of seconds (s) or minutes (min).
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Simplify (7/2 x 5/3) + (1/6 x 3/2) - (12/8 x 4/3)
Give proper step by step explanation
Answers
Answer:
To simplify the given expression:
(7/2 x 5/3) + (1/6 x 3/2) - (12/8 x 4/3)
Step 1: Simplify the fractions within the parentheses first.
(35/6) + (1/4) - (48/24)
Step 2: Find a common denominator for all three terms. The least common multiple of 6, 4, and 24 is 24.
(35/6 x 4/4) + (1/4 x 6/6) - (48/24 x 1/1)
Step 3: Simplify the numerators using the common denominator.
(140/24) + (6/24) - (48/24)
Step 4: Combine the like terms.
98/24 or 4 1/6
Therefore, the simplified form of the expression is 4 1/6.
Which postulate or theorem can be used to prove that ΔABC ≅ ΔDCB
Answers
The postulate or theorem that can be used to prove that ΔABC ≅ ΔDCB is the "Side-Side-Side (SSS) theorem".
Hence, the correct option is A.
Since in both triangles ΔABC and ΔDCB, we have
BC = BC (Common line)AB = CD (given)AC = BD (given)Therefore, by SSS theorem, we can conclude that ΔABC ≅ ΔDCB.
Hence, the correct option is A.
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(07.11a)marcus spent 10 hours doing his homework last week. this week he spent 11 hours doing homework. he says that he spent 110% more time doing homework this week. is he correct? show your work
Answers
Marcus is incorrect. The percentage increase in the time he spent doing homework this week compared to last week is 10%.
To determine if Marcus is correct, we need to calculate the percentage increase in the time he spent doing homework this week compared to last week.
First, we calculate the difference in hours:
11 hours - 10 hours = 1 hour
Then, we calculate the percentage increase:
(1 hour / 10 hours) x 100% = 10%
Therefore, Marcus is incorrect. The percentage increase in the time he spent doing homework this week compared to last week is 10%, not 110%.
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A dart has a circumference of 26\pi(pi symbol)calculate the total area on which the dart may land
Answers
The total area on which the dart may land is 530.93 square units
How to find the area of the dartThe dart is a circle and hence the calculations will be accomplished using formula pertaining to a circle.
The circumference of a circle (dart) is given by the formula below
C = 2 * π * r
where
C is the circumference
π = pi is a constant term and
r is the radius.
26π = 2πr
Dividing both sides by 2π, we get:
r = 13
Area of the dart (circle)
A = πr²
A = π * (13)²
A = 169π (in terms of pi)
A = 530.93 square units (to 2 decimal place)
Therefore, the total area on which the dart may land is 169π square units.
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Mike has some candies. he gave some to his friend. then, his mom gave him twice as much as he had in the beginning. how much did he have in the beginning if he has a total of 60 candies now?
Answers
According to given question Mike has 25 candies in the beginning.
Let's assume that Mike had "x" candies in the beginning.
After giving 15 candies to his friend, he would have had (x - 15) candies left.
His mom then bought him twice as many candies as he had in the beginning, which would be 2x candies.
So, the total number of candies Mike has now is (x - 15) + 2x = 60.
Combining like terms, we get 3x - 15 = 60.
Adding 15 to both sides, we have 3x = 75.
Finally, dividing both sides by 3, we find that x = 25.
Therefore, Mike had 25 candies in the beginning.
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The complete question is Mike had some candies. He gave 15 of them to his friend. After that, his mom bought him twice as many candies as he had in the beginning. How many candies did Mike have in the beginning if he now has a total of 60 candies?
Paulina plays both volleyball and soccer. The probability of her getting injured playing volleyball is 0. 10. 10, point, 1. The probability of her getting injured playing soccer is \dfrac{1}{10} 10
1
start fraction, 1, divided by, 10, end fraction
Answers
The probability of Paulina getting injured in either volleyball or soccer is 0.19.
To find the probability of Paulina getting injured in either volleyball or soccer, we can use the formula:
P(Volleyball or Soccer) = P(Volleyball) + P(Soccer) - P(Volleyball and Soccer)
We are given that the probability of Paulina getting injured playing volleyball is 0.1, and the probability of her getting injured playing soccer is 1/10 = 0.1 as well. However, we are not given any information about whether these events are independent or not, so we cannot assume that P(Volleyball and Soccer) is equal to the product of P(Volleyball) and P(Soccer).
If we assume that the events are independent, then we can calculate P(Volleyball and Soccer) as:
P(Volleyball and Soccer) = P(Volleyball) * P(Soccer) = 0.1 * 0.1 = 0.01
Then, using the formula above, we can calculate the probability of Paulina getting injured in either volleyball or soccer as:
P(Volleyball or Soccer) = 0.1 + 0.1 - 0.01 = 0.19
Therefore, the probability of Paulina getting injured in either volleyball or soccer is 0.19, assuming that the events are independent.
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Find the value of x
Hellpppp
Answers
Explanation is in the image!
Question
Select the correct answer from each drop-down menu. Nick bought apples at a farmers market where 5 apples cost $4. 45. If Nick bought 7 apples, he paid $. If Nick paid $9. 79 for some apples, he bought
apples.
Answers
Nick bought 11 apples for $9.79.
Nick paid $6.23 for 7 apples, and he bought 9 apples for $9.79.
Nick bought apples at a farmers market where 5 apples cost $4. 45. If Nick bought 7 apples, he paid $. To find the cost of one apple, we divide $4.45 by 5, which gives us $0.89 per apple.
For 7 apples, we multiply $0.89 by 7, which gives us $6.23.
To find the number of apples Nick bought for $9.79, we set up a proportion:
5 apples/$4.45 = x apples/$9.79
Solving for x, we get x = (9.79 × 5) / 4.45 ≈ 11
Therefore, Nick bought 11 apples for $9.79.
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A rectangle has vertices located at the points: A(-1 1/4,3 1/2), B(2 2/3,3 1/2), C(2 2/3,-1 3/4), D(-1 1/4,-1 3/4). Find the length of BC
Answers
The length of BC is 5.25 units. when the rectangle has vertices located at the points: B is ( 2 2/3,3 1/2) and C is (2 2/3,-1 3/4).
We need to find the length of BC. The length of a line segment BC can be calculated using the distance formula. The formula to find the distance between two points (x1, y1) and (x2, y2) is given as :
d = √(x2−x1)²+(y2−y1)²
Given data:
A = (-1 1/4,3 1/2)
B = ( 2 2/3,3 1/2)
C = (2 2/3,-1 3/4)
D = (-1 1/4,-1 3/4)
We need to Convert the B and C mixed numbers to improper fractions, we get,
B = (2+2/3), (3 + 1/2)
= (8/3, 7/2)
C = (2+2/3) , (-1 + 3/4)
= (8/3, -7/4).
Substituting the B and C values into the distance formula, we get:
= √(8/3 − 8/3)² + ( −7/4 − 7/2 )²
= √0 + ( − 21 / 4 )²
= √441/16
[tex]= 21/4[/tex]
[tex]= 5.25[/tex]
Therefore, the length of BC is 5.25 units.
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D. Larry has a new plan. Whenever his need for the car exceeds the number of slots he has rented the car for, he returns the car on time to avoid the fine, and uses his bicycle for the remaining deliveries. Unfortunately, bike deliveries take 3 times longer than car deliveries. Suppose he rents the car for 5 slots (75 minutes). If he needs 7 slots, then he uses the car for 5 slots, and the remaining 2 slots are done by bicycle, which take 6 slots (1. 5 hours). How much time does he expect to bicycle on average (in terms of slots)
Answers
Larry uses the car for the first 5 slots (75 minutes) and returns it on time to avoid any fine. He then uses his bike for the remaining 2 slots, which takes 6 slots (3 times longer than using the car).
Therefore, in total, Larry uses the car for 5 slots and the bike for 6 slots, for a total of 11 slots.
On average, Larry expects to use the bike for (6/11) of his total slots. Since each bike delivery takes 3 times longer than a car delivery, we can say that Larry expects to spend 3 times as much time on the bike as he would in the car. Therefore, the average time he expects to spend on the bike (in terms of slots) is:
(6/11) * 3 = 1.636 slots
So, Larry expects to spend an average of 1.636 slots (or about 24.5 minutes) on the bike for each delivery.
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Please help..... describe the transformation from the quadratic parent function f(x)=x^2
Answers
The quadratic parent function is f(x) = x^2, which is a U-shaped curve that passes through the origin. When we apply transformations to the quadratic parent function, its shape and position change accordingly.
One of the most common transformations applied to the quadratic parent function is vertical translation, which shifts the entire graph up or down. If we add a constant k to the function, the graph is shifted k units up. Similarly, if we subtract a constant k from the function, the graph is shifted k units down.Another common transformation is horizontal translation, which shifts the entire graph left or right.
If we replace x with x + h in the function, the graph is shifted h units to the left. If we replace x with x - h, the graph is shifted h units to the right.These transformations can be combined to create a variety of different quadratic functions. Each transformation changes the position or shape of the graph in a specific way, allowing us to create complex and interesting functions from the simple quadratic parent function.
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An art studio offers classes for painting and pottery. Each painting class is 1
hour long. Each pottery class is 1. 5 hours long. The art studio is only open
for classes a maximum of 40 hours per week, and only one class is offered at
a time. Each class costs $35, and the art studio earns a minimum of $1,000
per week from all classes. Let x be the number of painting classes offered per
week, and let y be the number of pottery classes offered per week.
Answers
The art studio can offer a maximum of 8 painting classes and 5 pottery classes per week, while still meeting the time constraint and earning at least $1000 per week.
To find the maximum number of classes the art studio can offer per week, we need to set up an equation based on the time constraint.
Let's assume that the studio offers x painting classes and y pottery classes per week. Since each painting class is 1 hour long and each pottery class is 1.5 hours long, the total time spent on classes can be represented by the equation:
1x + 1.5y ≤ 40
This equation states that the total number of hours spent on painting classes (1x) plus the total number of hours spent on pottery classes (1.5y) must be less than or equal to 40 hours per week.
To find the minimum revenue the art studio can earn per week, we can set up another equation based on the cost of each class and the minimum revenue requirement.
Let's assume that each painting or pottery class costs $35. Then the total revenue earned per week can be represented by the equation:
35x + 35y ≥ 1000
This equation states that the total revenue earned from painting classes (35x) plus the total revenue earned from pottery classes (35y) must be greater than or equal to $1000 per week.
Now we have two equations:
1x + 1.5y ≤ 40
35x + 35y ≥ 1000
We can use these equations to find the maximum number of classes the art studio can offer per week.
To do this, we can graph the two equations on the same coordinate plane and find the point where they intersect.
When we do this, we get the point (x, y) = (8, 16/3).
This means that the art studio can offer a maximum of 8 painting classes and 16/3 (or approximately 5.33) pottery classes per week, while still meeting the time constraint and earning at least $1000 per week.
Note that since the studio can only offer one class at a time, they would need to round down the number of pottery classes to 5 in order to offer a whole number of classes per week.
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Evaluate the following indefinite integral (1 / 2 +e^x + e^-x ) dx
Answers
We can rewrite the integrand as follows:
1 / (2 + e^x + e^(-x))
To evaluate this integral, we use the substitution u = e^x:
du/dx = e^x
dx = du/u
Substituting u and dx in terms of u into the integral, we get:
∫ (1 / (2 + u + 1/u)) du
Multiplying the numerator and denominator by u, we get:
∫ (u / (2u + u^2 + 1)) du
Next, we complete the square in the denominator:
u^2 + 2u + 1 = (u + 1)^2
So, we can write:
∫ (u / ((u + 1)^2 + 1)) du
Now, we use the substitution v = u + 1:
dv/du = 1
du = dv
Substituting v and du in terms of v into the integral, we get:
∫ ((v - 1) / (v^2 + 1)) dv
Using partial fractions, we can write:
(v - 1) / (v^2 + 1) = A(v - i) + B(v + i)
where A and B are constants to be determined, and i = sqrt(-1).
Multiplying both sides by v^2 + 1 and simplifying, we get:
v - 1 = A(v - i)(v^2 + 1) + B(v + i)(v^2 + 1)
Substituting v = i, we get:
-i - 1 = A(0) + B(2i)
B = -(i + 1)/2i = (1 - i)/2
Substituting v = -i, we get:
i - 1 = A(-2i) + B(0)
A = (1 - i)/2i = (i - 1)/2
Therefore, we have:
(v - 1) / (v^2 + 1) = [(i - 1)/(2i)](v + i) - [(1 - i)/(2i)](v - i)
Substituting back for v, we get:
(u / ((u + 1)^2 + 1)) = [(i - 1)/(2i)][(u + 1) + ie^x] - [(1 - i)/(2i)][(u + 1) - ie^x]
Substituting this expression back into the integral and simplifying, we get:
∫ (1 / (2 + e^x + e^(-x))) dx = [(i - 1)/(2i)]*ln(e^x + e^(-x) + 2) - [(1 - i)/(2i)]*ln(e^x - i) + C
where C is the constant of integration.
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A manufacturer of chocolate chips would like to know whether its bag filling machine works correctly at the 449 gram setting. It is believed that the machine is underfilling the bags. A 24 bag sample had a mean of 445 grams with a variance of 196. A level of significance of 0. 1 will be used. Assume the population distribution is approximately normal. Determine the decision rule for rejecting the null hypothesis. Round your answer to three decimal places
Answers
The decision rule for rejecting the null hypothesis in this scenario is to reject the null hypothesis if the sample mean falls below a critical value determined by the level of significance and the population parameters.
How to determine the decision rule for rejecting the null hypothesis in a chocolate chip bag filling machine test at the 449 gram setting?To determine the decision rule for rejecting the null hypothesis, we need to conduct a hypothesis test. In this case, the null hypothesis (H0) is that the bag filling machine works correctly at the 449 gram setting. The alternative hypothesis (Ha) is that the machine is underfilling the bags.
Since the sample size is 24 and the population variance is unknown, we can use the t-distribution for the hypothesis test. With a level of significance of 0.1 (or 10%), the critical t-value can be obtained from the t-distribution table.
Using the sample mean of 445 grams, the sample variance of 196, and the sample size of 24, we can calculate the t-value. The decision rule is to reject the null hypothesis if the calculated t-value is less than the critical t-value or greater than the negative of the critical t-value.
To obtain the specific critical t-value, we need the degrees of freedom, which is (sample size - 1). In this case, it is 24 - 1 = 23. Consulting the t-distribution table or using statistical software, we can find the critical t-value corresponding to a 10% significance level and 23 degrees of freedom.
Finally, we compare the calculated t-value to the critical t-value to determine whether to reject the null hypothesis or not.
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For the function f(x) = x^2e^7x find the critical numbers and use the First Derivative Test to find the local
maximum and minimum values and where they occur. Enter exact answers in fraction form.
Answers
For the function f(x) = x^2e^7x the critical numbers are 0 and -2/7 and using the First Derivative Test local minima at x = 0 and -2/7.
The function f(x) = x²e⁷ˣ
Differentiate the function f(x) with respect to x
f'(x) = 2xe⁷ˣ + 7x²e⁷ˣ
f'(x) = xe⁷ˣ(2 + 7x)
Now put f'(x) = 0 for critical points.
xe⁷ˣ(2 + 7x) = 0
x = 0 2 + 7x = 0 e⁷ˣ ≠ 0
x = 0 x = -2/7 e⁷ˣ ≠ 0
Hence, 0 and -2/7 are the critical points.
f'(x) = 2xe⁷ˣ + 14x²e⁷ˣ
Now f''(x) = 2e⁷ˣ + 14xe⁷ˣ + 28xe⁷ˣ + 98x²e⁷ˣ
f''(x) = 2e⁷ˣ + 42xe⁷ˣ + 98x²e⁷ˣ
f''(x) = 2e⁷ˣ(1 + 21x + 49x²)
Now f''(x) at x = 0
f''(0) = 2e⁷⁽⁰⁾(1 + 21(0) + 49(0)²)
f''(0) = 2(1 + 0 + 0)
f''(0) = 2 > 0
So f(x) has minima at x = 0
Local minima at x = 0 and y = 0.
Now f''(x) at x = -2/7
f''(0) = 2e⁻²(1 + 21(-2/7) + 49(-2/7)²)
f''(0) = 2e⁻²(1 - 6 + 4)
f''(0) = -2e⁻² < 0
f(x) has local minima at x = -2/7
Now y at x = -2/7 is
y = f(-2/7)
y = (-2/7)² e⁻²
y = 4/49e²
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How many numbers from 2 through 100 can be expressed as p², where p is a prime number?
F. 4
G. 5
H. 7
J. 8
K. 9
