The given statement "A function f(x) = 3x^4 dominates g(x) = x^4" is True, which means that as x gets larger, the value of f(x) will increase much more rapidly than the value of g(x).
As x increases or decreases, the 3x^4 term in f(x) will grow faster or be larger in magnitude than the x^4 term in g(x). Since f(x) grows faster or has a larger magnitude than g(x), we can conclude that f(x) dominates g(x).
Therefore, the function f(x) = 3x^4 has a higher degree than g(x) = x^4
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A movie studio surveyed married couples about the types of movies they prefer. In the survey, the husband and wife were each asked if they prefer action, comedy, or drama. The summary of the data the studio got after asking 225 couples
Suppose the movie studio will ask 150 more couples about their movie preference. How many of these 150 couples will have exactly one spouse prefer action movie?
Out of the 150 new couples, we can expect about:
45 * (150/240) = 28.125 couples where the husband prefers action but the wife does not.
30 * (150/240) = 18.75 couples where the wife prefers action but the husband does not.
What is probability?
Probability is a measure of the likelihood of an event occurring.
Based on the given data from the survey of 225 couples, we can construct a contingency table as follows:
Husband Wife Total
Action 45 30 75
Comedy 30 45 75
Drama 45 45 90
Total 120 120 240
From the contingency table, we can see that:
Out of 240 respondents, 75 (45 from husbands and 30 from wives) preferred action movies.
Out of 240 respondents, 60 (30 from husbands and 30 from wives) preferred comedy movies.
Out of 240 respondents, 90 (45 from husbands and 45 from wives) preferred drama movies.
To answer the question of how many of the 150 couples will have exactly one spouse who prefers action movie, we can use the information that:
Out of 240 respondents, 45 husbands preferred action movies but their wives did not.
Out of 240 respondents, 30 wives preferred action movies but their husbands did not.
Therefore, out of the 150 new couples, we can expect about:
45 * (150/240) = 28.125 couples where the husband prefers action but the wife does not.
30 * (150/240) = 18.75 couples where the wife prefers action but the husband does not.
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Chris bought 5 tacos and 2 burritos for $13. 25.
Brett bought 3 tacos and 2 burritos for $10. 75.
The price of one taco is $
The price of one burrito is $
If Chris bought 5 tacos and 2 burritos for $13. 25 and Brett bought 3 tacos and 2 burritos for $10. 75, the price of one taco is $1.25, and the price of one burrito is $3.50.
Let the price of one taco be T and the price of one burrito be B. We have the following equations:
5T + 2B = $13.25
3T + 2B = $10.75
To find the prices of the taco and the burrito, we can use the system of equations. First, subtract the second equation from the first equation:
(5T + 2B) - (3T + 2B) = $13.25 - $10.75
2T = $2.50
Now, divide by 2 to find the price of one taco:
T = $1.25
Next, plug the value of T back into one of the equations (let's use the second equation):
3($1.25) + 2B = $10.75
$3.75 + 2B = $10.75
Now, subtract $3.75 from both sides:
2B = $7.00
Finally, divide by 2 to find the price of one burrito:
B = $3.50
So, the price of one taco is $1.25, and the price of one burrito is $3.50.
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Evaluate the integral dy (tan-'[y/8)) (64+y?) ( 1 + dy (tan-'(4/8)) (64+y?) =
Answer: ln|y/8| + C
Explanation:
First, we need to recognize that the derivative of arctan(x) is 1/(1+x^2). Therefore, the derivative of arctan(y/8) is 8/(64+y^2).
Now, using the substitution u = y/8, we can rewrite the integral as:
∫(1/u)(64+64u^2)(8/(64+64u^2))du
Simplifying, we get:
∫(1/u)du = ln|u| = ln|y/8|
Therefore, the final answer is:
ln|y/8| + C
where C is the constant of integration.
a) by using Venn-diagram. 75 students in a class like picnic or hiking or both. Out of them 10 like both the activities. The ratio of the number of students who like picnic to those who like hiking is 2 : 3. (i) Represent the above information in a Venn-diagram. (ii) Find the number of students who like picnic. (iii) Find the number of students who like hiking only. (iv) Find the percentage of students who like picnic only.
(i) A Venn-diagram of this information is shown below.
(ii) The number of students who like picnic = 34
(iii) The number of students who like hiking only = 51
(iv) The percentage of students who like picnic only. = 45.33%
Let us assume that A represents the set of students who like picnic.
B represents the set of students who like the hiking.
The total number of students in a class are: n(A U B) = 75
Out of 75 students, 10 like both the activities.
n(A ∩ B) = 10
The ratio of the number of students who like picnic to those who like hiking is 2 : 3
Let number of students like tea n(A) = 2x
and the number of students like coffee n(B) = 3x
n(A U B) = n(A) + n(B) - n(A ∩ B)
75 = 2x + 3x - 10
75 + 10 = 5x
85/5= x
x = 17
The number of students like picnic = 2x
= 2 × 17
= 34
The number of students like hiking = 3x
= 3 × 17
= 51
This informtaion in Venn diagram is shown below.
The percentage of students who like picnic only would be,
(34/75) × 100 = 45.33%
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CAN SOMEONE SHOW ME STEP BY STEP ON HOW TO DO THIS
A city just opened a new playground for children in the community. An image of the land that the playground is on is shown.
A polygon with a horizontal top side labeled 45 yards. The left vertical side is 20 yards. There is a dashed vertical line segment drawn from the right vertex of the top to the bottom right vertex. There is a dashed horizontal line from the bottom left vertex to the dashed vertical, leaving the length from that intersection to the bottom right vertex as 10 yards. There is another dashed horizontal line that comes from the vertex on the right that intersects the vertical dashed line, and it is labeled 12 yards.
What is the area of the playground?
900 square yards
855 square yards
1,710 square yards
The volume of this cube is 19,683 cubic yards. What is the value of s?
The value of s is, 27 yards
:: Volume of cube with side s, is equal to s³
So, as the given volume is 19,683 cubic yards.
Therefore, it can related as,
s³ = 19,683 (yards)³
So,
s = ∛(19,683) yards
s = 27 yards
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A linear function that represents the number of animals adopted from the shelter is compared to a different linear function that represents the hours volunteers work at the shelter each week. describe the key features of the functions that are needed to determine if these lines intersect.
please help i don't understand >.
To determine if two lines intersect, compare their slopes and y-intercepts, and solve their equations simultaneously.
How to determine if two lines intersect?To determine if two lines intersect, you need to compare their key features, such as their slope and y-intercept.
If the slopes of the two lines are different, then they will intersect at some point.
If the slopes are the same, then the lines may or may not intersect, depending on whether or not their y-intercepts are also the same.
To find the point of intersection, you can set the two linear functions equal to each other and solve for the variable. The resulting value will give you the x-coordinate of the intersection point, which can then be substituted back into either equation to find the corresponding y-coordinate.
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Brooke and eileen are working on a math problem together and can't seem to agree on an answer. their teacher drew this number line on the board and asked them to think of a situation that could be represented by it.
brooke suggested the following situation:
christopher wants to buy a new bicycle and needs to earn more than $75 in order to have enough money.
eileen suggested the following situation:
paul is flying home from vacation and has less than 75 minutes left of the flight.
Both situations can be represented by the number line as they both involve values either greater than or less than 75.
The number line the teacher drew can represent both Brooke's and Eileen's situations.
In Brooke's situation, the number line can represent the amount of money Christopher needs to earn to buy a new bicycle. If he needs to earn more than $75, any point on the number line greater than 75 would represent the amount of money he has earned that is sufficient for purchasing the bicycle.
In Eileen's situation, the number line can represent the time left in Paul's flight. If Paul has less than 75 minutes left, any point on the number line less than 75 would represent the time remaining in his flight.
Both situations can be represented by the number line as they both involve values either greater than or less than 75.
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A project is budgeted for 1,200 hours and will last 6 weeks. a technician will be 25% billable to the project for the first three weeks and then 100% for the final three weeks. if a technician normally works 40 hours per week, how many total hours will the technician bill to the job?
The technician will bill a total of 150 hours to the job. To find the total hours, we need to determine how many hours the technician will work during the first three weeks and the last three weeks, and then add them together.
1. First three weeks:
The technician will be 25% billable during these weeks. They work 40 hours per week, so we need to calculate 25% of 40 hours for each week:
25% of 40 hours = 0.25 * 40 = 10 hours per week
Since there are three weeks, we'll multiply these hours by 3:
10 hours/week * 3 weeks = 30 hours
2. Last three weeks:
The technician will be 100% billable during these weeks. They work 40 hours per week, so they'll bill 40 hours for each of these weeks:
40 hours/week * 3 weeks = 120 hours
3. Finally,
we need to add the hours from the first and last three weeks together to find the total hours the technician will bill to the job:
30 hours (first three weeks) + 120 hours (last three weeks) = 150 hours
The technician will bill a total of 150 hours to the job.
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Determine the distance between the points (−3, −6) and (5, 0).
[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{-3}~,~\stackrel{y_1}{-6})\qquad (\stackrel{x_2}{5}~,~\stackrel{y_2}{0})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{(~~5 - (-3)~~)^2 + (~~0 - (-6)~~)^2} \implies d=\sqrt{(5 +3)^2 + (0 +6)^2} \\\\\\ d=\sqrt{( 8 )^2 + ( 6 )^2} \implies d=\sqrt{ 64 + 36 } \implies d=\sqrt{ 100 }\implies d=10[/tex]
HELPPP JUST 1 QUESTION!!! QUESTION IN PICTURE
Answer:
48.91
Step-by-step explanation:
r=cos^-1(.92)
r=23.07
cos(23.07)=45/y
y=45/cos(23.07)
48.91
what is the answer??
The equation that could represent each of the graphed polynomial function include the following:
First graph: y = x(x + 2)(x - 3)
Second graph: y = x⁴ - 5x² + 4
What is a polynomial graph?In Mathematics and Geometry, a polynomial graph simply refers to a type of graph that touches the x-axis at zeros, roots, solutions, x-intercepts, and factors with even multiplicities.
Generally speaking, the zero of a polynomial function simply refers to a point where it crosses or cuts the x-axis of a graph.
By critically observing the graph of the polynomial function shown in the image attached above, we can logically deduce that the first graph has a zero of multiplicity 1 at x = 2 and zero of multiplicity 1 at x = -3.
Similarly, we can logically deduce that the second graph has a zero of multiplicity 2 at x = 2 and zero of multiplicity 2 at x = -2.
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AREA OF TRAPEZOID PLS ANSWER ASAP
Which of these variables is your dependent variable?
How many jumps I can do
Which one is the independent variable?
How long I am jumping (2 minutes)
Write a sentence that describes the relationship between the dependent variable and the independent variable. (Hint: Ratio language can help. )
In this scenario, the dependent variable is "how many jumps I can do," while the independent variable is "how long I am jumping (2 minutes)."
The relationship between these variables can be described as follows: The number of jumps completed depends on the duration of time spent jumping, with a specific focus on a 2-minute interval.
When we say the dependent variable is "how many jumps I can do," it means that the number of jumps completed is determined by or depends on the independent variable, which is the duration of time spent jumping.
This suggests that as the duration of time increases or decreases, it will likely have an impact on the number of jumps performed.
In this particular case, you have specified a 2-minute interval as the focus. It suggests that you are examining the relationship between the number of jumps completed and the specific duration of 2 minutes.
This implies that you are interested in understanding how the number of jumps varies within this fixed time frame.
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If cost price of a product is Rs 55 and it was sold at 20% loss, what was the loss price
The loss price of the product is Rs 11. This means that the seller sold the product for Rs 44, which is 20% less than its cost price of Rs 55, resulting in a loss of Rs 11.
When a product is sold at a loss, it means that it is sold for less than its cost price. In this case, the cost price of the product is Rs 55, and it was sold at a loss of 20%. This means that the selling price of the product is 80% of its cost price. To find out the selling price, we can multiply the cost price by 80% or 0.8.
Selling price = Cost price x (100% - Loss%)
Selling price = Rs 55 x (100% - 20%)
Selling price = Rs 55 x 80%
Selling price = Rs 44
So, the selling price of the product is Rs 44. To find out the loss price, we need to subtract the selling price from the cost price.
Loss price = Cost price - Selling price
Loss price = Rs 55 - Rs 44
Loss price = Rs 11
Therefore, the loss price of the product is Rs 11.
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can someone pls help with this
A linear function would be the best fit for the data.
A function that would be the best for this data is: D. y = -4/25(x) + 10
The amount of snow that would be on the ground when the temperature reaches 55° is 1.2 inches.
How to determine the line of best fit?In this scenario, the temperature would be plotted on the x-axis (x-coordinate) of the scatter plot while the snow (inches) would be plotted on the y-axis (y-coordinate) of the scatter plot through the use of Microsoft Excel.
On the Microsoft Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an equation for the line of best fit (trend line) on the scatter plot.
From the scatter plot (see attachment) which models the relationship between the temperature and the snow (inches), an equation for the line of best fit is given by:
y = -0.16x + 10
y = -4/25(x) + 10
When x = 55, the amount of snow is given by;
y = -4/25(55) + 10
y = 1.2 inches.
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A movie theater has a seating capacity of 349. The theater charges $5. 00 for children, $7. 00 for students, and $12. 00 for adults. There are half as many adults as there are children. If the total ticket sales was $ 2540, How many children, students, and adults attended?
194 children, 58 students, and 97 adults attended the movie.
Let's use algebra to solve this problem.
Let's assume the number of children who attended the movie is C, the number of students is S, and the number of adults is A.
From the problem, we know that:
The seating capacity of the theater is 349:
C + S + A = 349
The theater charges $5 for children, $7 for students, and $12 for adults:
5C + 7S + 12A = $2540
There are half as many adults as there are children:
A = 1/2C
Now we can substitute A = 1/2C from the third equation into the first and second equations:
C + S + 1/2C = 349
3/2C + S = 349
5C + 7S + 12(1/2C) = $2540
5C + 7S + 6C = $2540
11C + 7S = $2540
Now we have two equations with two variables, C and S.
We can solve for S in the first equation:
3/2C + S = 349
S = 349 - 3/2C
Now we can substitute S = 349 - 3/2C into the second equation:
11C + 7S = $2540
11C + 7(349 - 3/2C) = $2540
11C + 2443 - 10.5C = $2540
0.5C = 97
C = 194
Therefore, 194 children attended the movie of total sales.
We can use A = 1/2C from the third equation to find the number of adults:
A = 1/2C
A = 1/2(194)
A = 97
Therefore, 97 adults attended the movie.
We can use C + S + A = 349 to find the number of students:
C + S + A = 349
194 + S + 97 = 349
S = 58
Therefore, 58 students attended the movie.
In summary, 194 children, 58 students, and 97 adults attended the movie.
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the box-and-whisker plot shows the number of pigeons spotted by visitors at the park during the last weekend. the horizontal axis ranges from 0 to 20 in increments of 1. a horizontal line segment, or whisker, begins at 1 and ends on the left vertical side of the rectangle at 8. a vertical line segment passes through the rectangle at 10. the right vertical side of the rectangle is at 11. a second horizontal line segment, or whisker, begins on the right vertical side of the rectangle and ends at 13. what is the range of the data?
The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1) of the data. From the box-and-whisker plot given, the IQR is 12.
The box-and-whisker plot provides us with the following information:
The minimum value is 1 (the left end of the left whisker)The first quartile (Q1) is 8 (the end of the left whisker)The median (Q2) is 10 (the middle of the box)The third quartile (Q3) is 11 (the end of the right whisker)The maximum value is 13 (the right end of the right whisker)Therefore, the range of the data is the difference between the maximum and minimum values:
Range = maximum value - minimum value = 13 - 1 = 12
So, the range of the data is 12.
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Full Question: The box-and-whisker plot shows the number of pigeons spotted by visitors at the park during the last weekend. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 What is the interquartile range of the data? Provide your answer below:
Image attached
11. The volume of a cuboid with a square base is given 5 by (2x¹ + xy-2y) m². 5 (i) Factorise the expression 2x² + xy-2y². 1 (ii) The cuboid has a height of m. Given that the length of each side of the base can be expressed as (px - qy) m or (qx + py) m, using your answer from part (i), state the value of p and of q. (iii) Hence, express x in terms of y.
Catering company provides packages for weddings and for showers. The cost per person for small groups is
pproximately Normally distributed for both weddings and showers. The mean cost for weddings is $82. 30 with a
andard deviation of $18. 20, while the mean cost for showers is $65 with a standard deviation of $17. 73. If 9
eddings and 6 showers are randomly selected, what is the probability the mean cost of the weddings is more than
e mean cost of the showers?
The probability that the mean cost of the 9 weddings is more than the mean cost of the 6 showers is approximately 0.0207 or 2.07%.
The probability that the mean cost of the 9 weddings is more than the mean cost of the 6 showers can be found using the Z-score and the difference between the means of two normally distributed variables.
1: Calculate the difference in means and standard deviations.
Δμ = μ_weddings - μ_showers = $82.30 - $65 = $17.30
Δσ = sqrt((σ_weddings²/n_weddings) + (σ_showers²/n_showers)) = sqrt((18.20²/9) + (17.73²/6)) = $8.47
2: Calculate the Z-score.
Z = (Δμ - 0) / Δσ = (17.30 - 0) / 8.47 ≈ 2.04
3: Determine the probability using a Z-table.
P(Z > 2.04) ≈ 0.0207
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6. Quadrilateral ABCD is dilated with center C and a scale factor of 1/2.Draw A'B'C'D'.
Thus, the coordinates of Quadrilateral A'B'C'D' after the dilation with the scale factor of 1/2 are - A'(1.5, 2), B'(0.5, 5), C'(6, 7), D'(4.5, 1.4).
Explain about the dilation:In geometry, a dilation is a transformation that alters an object's size without altering its general shape.
If the dilation factor is greater than 1, the item grows in size. The size shrinks .if the factor is between 0 and 1, and such dilations are occasionally referred to as compressions.Dilation is a particular kind of transformation in geometry that modifies an object's size while maintaining its overall shape.
Given:
scale factor = 1/2
coordinates of Quadrilateral ABCD
A(3,4) , B(1,10) ,C(12,14), D(9,3)
Now, coordinates about the dilation with centre C:, multiply each coordinate with 1/2.
A'(3*1/2,4*1/2) --> A'(1.5, 2)
B'(1*1/2,10*1/2) ---> B'(0.5, 5)
C'(12*1/2,14*1/2), --> C'(6, 7)
D'(9*1/2,3*1/2) ---> D'(4.5, 1.4)
Thus, the coordinates of Quadrilateral A'B'C'D' after the dilation with the scale factor of 1/2 are - A'(1.5, 2), B'(0.5, 5), C'(6, 7), D'(4.5, 1.4).
Graph is attached.
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This graph represents the equation y=(x-5)^2-1 .
How many ordered pairs (x, y) for 3 < x < 7 satisfy this equation?
There are 3 ordered pairs (x, y) that satisfy the equation y=(x-5)^2-1.
To find the ordered pairs (x, y) for 3 < x < 7 that satisfy the equation y=(x-5)^2-1, follow these steps:
Step 1: Set the range of x values: 3 < x < 7
Step 2: Plug in each whole number value of x within the given range (4, 5, and 6) into the equation and calculate the corresponding y values.
For x = 4:
y = (4 - 5)^2 - 1
y = (-1)^2 - 1
y = 0
For x = 5:
y = (5 - 5)^2 - 1
y = (0)^2 - 1
y = -1
For x = 6:
y = (6 - 5)^2 - 1
y = (1)^2 - 1
y = 0
Step 3: Write the ordered pairs (x, y) based on the calculated y values.
For x = 4, the ordered pair is (4, 0)
For x = 5, the ordered pair is (5, -1)
For x = 6, the ordered pair is (6, 0)
In the given range, there are 3 ordered pairs (x, y) that satisfy the equation y=(x-5)^2-1.
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A pond of fish starts with 200 fish. The pond can sustain 460 fish, 40% of the fish die each year while the number of births is 60% of the current population. – 3.04174E+07 fish are harvested from the pond each year. Write a differential equation that models the problem
The differential equation that models the problem is: dN/dt = 0.2*N(t) - 3.04174E+07.
Let's denote the current number of fish in the pond by N(t), where t is time in years.
The rate of change of N(t) is given by the difference between the number of births and deaths, minus the number of fish harvested from the pond:
dN/dt = (0.6N(t)) - (0.4N(t)) - (3.04174E+07)
The first term represents the number of births, which is 60% of the current population N(t). The second term represents the number of deaths, which is 40% of the current population N(t). The third term represents the number of fish harvested from the pond each year.
Therefore, the differential equation that models the problem is:
dN/dt = 0.2*N(t) - 3.04174E+07
Note that we have simplified the expression (0.6-0.4)N(t) to 0.2*N(t) for simplicity.
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How can I get the answer for
A=
Vertex for y=
Answer:
1) a = 14
2) -4 (x - 2)² - 5
Step-by-step explanation:
To obtain a vertex, you take h and k in a equation.
So a(x-h)²+k = a(x-2)² -5
For the point (1, - 9),
a[(1)-2]² - 5 = - 9
a(1) = -9+5
a = -4
so the final equation is
-4(x-2)² - 5
I'm not 100% sure about this but I tried. Let me know if it makes sense
A triangle has sides of length 12, 17, and 22. of the measures of the three interior angles, which is the greatest of the three
The greatest of the three interior angles in the triangle is approximately 71.2 degrees.
To find out which of the three interior angles in the triangle is the greatest, we can use the fact that the largest angle is opposite the longest side. So, in this case, the longest side is 22, which means that the angle opposite it must be the greatest. We can use the Law of Cosines to find the measure of this angle:
c^2 = a^2 + b^2 - 2ab*cos(C)
where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides. Plugging in the values we know, we get:
22^2 = 12^2 + 17^2 - 2*12*17*cos(C)
484 = 144 + 289 - 408*cos(C)
51 = 408*cos(C)
cos(C) = 51/408
C = cos^-1(51/408)
C ≈ 71.2 degrees
So the greatest of the three interior angles in the triangle is approximately 71.2 degrees.
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Let w = 2xy + y2 - 4x2, += st, y=,= Compute Bu (1.-3) - 88 -(1, -3)
To compute Bu(1.-3) - 88 - (1, -3), we need to substitute the values of u and v into the expression for w.
First, we need to find the values of u and v. Since u = 1.-3 and v = (1, -3), we have:
u = 1.-3 = 1 - 0.3 = 0.7
v = (1, -3)
Next, we can substitute these values into the expression for w:
w = 2xy + y^2 - 4x^2
= 2(1)(-3) + (-3)^2 - 4(1)^2 (substituting x = 1 and y = -3)
= -6 + 9 - 4
= -1
Finally, we can compute Bu(1.-3) - 88 - (1, -3) by multiplying the gradient of w by the vector (1, -3) and subtracting 88:
Bu(1.-3) - 88 - (1, -3) = (-8x + 2y, 2x + 2y) (1, -3) - 88
= (-8(1) + 2(-3), 2(1) + 2(-3)) (1, -3) - 88
= (-14, -4) (1, -3) - 88
= (-14)(1) + (-4)(-3) - 88
= -14 + 12 - 88
= -90
Therefore, Bu(1.-3) - 88 - (1, -3) = -90.
Since the question seems to have some typos or missing information, I'll assume you want to find the partial derivatives of w with respect to x and y, and evaluate them at the point (1, -3).
Given w = 2xy + y² - 4x², let's compute the partial derivatives:
∂w/∂x = 2y - 8x
∂w/∂y = 2x + 2y
Now, let's evaluate these partial derivatives at the point (1, -3):
∂w/∂x(1, -3) = 2(-3) - 8(1) = -6 - 8 = -14
∂w/∂y(1, -3) = 2(1) + 2(-3) = 2 - 6 = -4
Thus, the evaluated partial derivatives are ∂w/∂x(1, -3) = -14 and ∂w/∂y(1, -3) = -4.
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The value V of a classic car
appreciates exponentially and is
represented by V = 32,000(1.18)t
,
where t is the number of years
since 2020.
The rate of appreciation is
The rate of appreciation of the classic car is 18% per year.
Define exponentAn exponent is a mathematical operation that indicates how many times a number or expression is multiplied by itself. It is represented by a superscript number that is written to the right and above the base number or expression. The exponent tells us how many times the base is multiplied by itself.
The value V of the classic car appreciates exponentially, and it is represented by the formula:
V = 32,000[tex]1.18^{2}[/tex]
The term [tex]1.18^{t}[/tex] represents the factor by which the value of the car increases each year. If we calculate this factor for one year (t=1), we get:
(1.18)¹= 1.18
This means that the value of the car increases by 18% in the first year. Similarly, if we calculate the factor for two years (t=2), we get:
(1.18)² = 1.39
This means that the value of the car increases by 39% in the first two years (18% in the first year and an additional 21% in the second year).
Therefore, the rate of appreciation of the classic car is 18% per year.
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Which of the following combinations of side lengths would NOT form a triangle with vertices X, Y, and Z?
A.
XY = 7 mm , YZ = 14 mm , XZ = 25 mm
B.
XY = 11 mm , YZ = 18 mm , XZ = 21 mm
C.
XY = 11 mm , YZ = 14 mm , XZ = 21 mm
D.
XY = 7 mm , YZ = 14 mm , XZ = 17 mm
The combination of side lengths that would not form a triangle is C.XY = 11 mm, YZ = 14 mm, XZ = 21 mm.
We shall use the triangle inequality theorem to determine if a set of side lengths can form a triangle.
What is the triangle inequality theorem?The triangle inequality theorem says that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
We shall calculate each of the options:
For option A:
XY + YZ = 7 mm + 14 mm = 21 mm which is < XZ = 25 mm.
Therefore, option A does form a triangle.
For option B:
XY + YZ = 11 mm + 18 mm = 29 mm, which is > XZ = 21 mm.
YZ + XZ = 18 mm + 21 mm = 39 mm, which is > XY = 11 mm.
XY + XZ = 11 mm + 21 mm = 32 mm, which is > YZ = 18 mm.
Therefore, option B does form a triangle.
For option C:
XY + YZ = 11 mm + 14 mm = 25 mm, and is > XZ = 21 mm.
Therefore, option C does not form a triangle.
For option D:
XY + YZ = 7 mm + 14 mm = 21 mm, which is > XZ = 17 mm.
YZ + XZ = 14 mm + 17 mm = 31 mm, which is > XY = 7 mm.
XY + XZ = 7 mm + 17 mm = 24 mm, which is > YZ = 14 mm.
Therefore, option D does form a triangle.
Therefore, the combination of side lengths that would not form a triangle is XY = 11 mm, YZ = 14 mm, XZ = 21 mm.
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Peter creates a square pyramid
model for History class. The
base of the pyramid has an
area of 20 square inches. Each
triangle has an area of 10
square inches. If Peter wants to
cover the entire pyramid with
gold paper, how much paper
will he need?
Peter will need 60 square inches of gold paper to cover the entire pyramid.
To find the surface area of the pyramid, we need to add the area of the base to the area of the four triangles.
Area of the base = 20 square inches
Area of each triangle = 10 square inches
Total area of the four triangles = 4 x 10 = 40 square inches
Total surface area = Area of base + Total area of four triangles
= 20 + 40
= 60 square inches
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A toy tugboat is launched from the side of a pond and travels North at 5cm/s. At the same moment, a toy sail ship from a point 8sqrt(2) m. Northeast of the tugboat and travels West at 7 cm/s. How closely do the two toys approach each other?\
The toys approach each other at the distance of 630 cm.
To solve the problem, we can use the Pythagorean theorem.
Let the distance between the tugboat and the sail ship be d, and
let t be the time in seconds since they started moving.
Then we have:
Distance traveled by the tugboat (in cm) = 5t
Distance traveled by the sail ship (in cm) = 7t/sqrt(2)
Using the Pythagorean theorem, we have:
d² = (5t)² + (7t/(\sqrt(2)))²
d² = 25t² + 24.5t²
d² = 49.5t²
d = \sqrt(49.5)t
To find how closely the two toys approach each other, we need to find the minimum value of d.
This occurs when t is maximized, which happens when the toys are closest to each other.
The sail ship travels a distance of 8\sqrt(2) meters in the Northeast direction, which is equivalent to 800\sqrt(2) cm. Therefore, the time taken for the sail ship to travel this distance is:
t = (800\sqrt(2) cm) / (7 cm/(\sqrt(2))) = 200\sqrt(2) seconds
Substituting this value of t in the equation for d, we get:
d = \sqrt(49.5)(200\sqrt(2)) = 630 cm (corrected)
Therefore, the minimum distance between the two toys is 630 cm.
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